I teach my students that two of the most powerful tricks in Algebra are adding zero and multiplying by 1. When you complete the square you add zero, and when you add fractions you are multiplying each fraction by 1. In a proof class you have to do this more often and more creatively. It's similar with geometry. You have to create. But this time instead of creating a clever 1 or 0, you are creating a line or other geometric entity that expands your diagram. It's just a different form of creativity. It comes naturally to some people and less naturally to others.
One of the reasons why geometry is one of my favourite areas of Mathematics is because of the thought that it requires. You have to look at things outside of the box (no pun intended) and you are rewarded for your creativity. Due to this, there are an endless number of possible solutions to just one geometry question.
Great post. Geometry, in my opinion, is the most important subject in high-school. Best class to develop pure conceptual problem solving. Applies to all parts of education/life.
@@heatmyzer9 It saddens me when people say stuff like "I'm not even gonna need to do maths in my future career so there's no point of even paying attention." Maths is so much more than knowing how to crunch numbers, it also MASSIVELY develops critical thinking and creativity.
@@staticchimera44 Agree 100%. Life/success involves solving all kinds of problems and making decisions. If you can’t do those two things….your path through life is greatly narrowed. 0.01% direct use of geometry by most….99% indirect application through creative problem solving.
@@staticchimera44 agree. The math problems might not be useful in your daily life but knowing how to solve problems gets way better by solving such problems
Speaking about the difficulty of synthetic geometry, I would love you to make a video about using complex numbers in geo problems. I'm sure a lot of people don't know about that topic.
Thanks for saying this! I watch quite a few geometry videos on TH-cam, and I'm always asking myself how you're supposed to know where to start. Most videos are like, "now do this random arbitrary thing that just lets you solve the problem," as if it's obvious what to do.
My friend gave this particular qn to me 2-3 yrs ago. And I was able to solve it in 1 period(40 mins) in school but got into trouble getting the notes what the teacjer had done during the class.
Finally, someone who thinks geometry is hard too...and yes, exactly, you gotta be very observative to know how to do the problems...when I was at 7th grade I loved geometry because it was quite easy...at 8th grade I started to hate it a little bit because it was harder, it required more observation to do it...same thing at 9th grade, I hated it even more, now I finished 10th grade and what i can say, I love algebra so much...and I already watched some of your videos with calculus, which I should study in my next 2 years, and I learned a lot from those videos, thanks to you
I completely understand what you say. I have lived the same and then later I came to love those challenges. And that is why I can and should advice you to learn to not hate what challenges you, but take the challenge as an oportunity to grow and improve. Otherwise nothing of value can be achieved. THE KEY TO MAKE IT is to start where you are not strugling a lot but just a little, then from there progress with a pace that is still comfortable but not far from too fast with the right guidance of a good teacher, tutor or coach. Have fun, make friends in those batles, do it for the fun or exploring and the thrill of discovering, for the joy of intelectual growth. Enjoy!
Didficult geometry problems always take way more time to solve than equally difficult algebra/precalc problems. I mean I remember we had some olympid level problems given to us to solve at home and I ended up just sitting and looking at the god damn drawing for hours.
I agree with you sir from my experience as an IMO participant. Solutions that depend only on Euclidean geometry are really nice but really tough to discover. How in the world can we know that we need to construct this line, draw this circle, or even define new points?! Luckily there are other tools that made my life easier (Trigonometry - complex coordinates - barycentric coordinates - projective geometry). I have discussed these useful tools in short in one of my videos about what topics one should study to prepare for math Olympiad contest. I also have just started a geometry tutorial on my channel! So yes Geo is hard, but that's exactly what makes it beautiful 😉
Being very used to algebra really make reasoning in geometric terms really hard, at least for me. That's why reading through Euclid's Element is harder for me than reading through calculus books. What blew my mind was that the ancient greek used geometric reasoning for everything, that's how we get the terms like square of a number, completing the squre and so on. Don't even get me started on Appolonius' Conics, it literally melted my brain.
I'd say that the argument "how in the world do I know to do it" isn't specific to geometry. Just review your own algebra/calculus videos to see you're doing tricks nobody without experience will come up with. Looking at your "my first quintic" or "a brilliant limit"
No, in algebra, you don’t even need the actual slick choices (even numerical methods and computational) You just need the option of having a slick choice
@@duckymomo7935 That is true. Algebra you can use brute force, take a long as time but it can be done. For example, expanding a binominal. You can either know the coefficients and powers, or you can literally sit there multiplying out. It's ugly as hell, but doable.
@@Tony29103 Disagree. It's not all about powers and binomials in algebra. There are problems which will make even experienced people stuck. Examples - equations like these: √(x-1/𝑥)+√(1-1/𝑥)=𝑥 ∛(𝑥+22)=𝑥³+6𝑥²+12𝑥-32 𝑥+𝑥/√(𝑥²-1)=35/12 etc; for such you'll need to know a trick or sometimes a combination of right methods (and still stumble for a while) - and good luck "brute-forcing" those with raising powers, that's lead you nowhere close to solving them.
I like to prove the star one thinking about the external angles of the pentagon and the 5 triangles. Because actually, the angles a,b,c,d and e are the only ones that aren't from the external angles of the pentagon. So we have that the sum of the 15 angles from the triangles must be 900⁰, but the 10 angles from the bases of the triangles are actually 2 times the external angles of the pentagon, so they are 2×360⁰=720⁰, then we have 720⁰+a+b+c+d+e=900⁰, a+b+c+d+e=180⁰
To be honest the "star" problem isn't hard... There's a simple solution using exterior angle of triangle where you will obtain a triangle with the angles equal to b, a+d, c+e hence a+b+c+d+e=180 degrees.
There is also a more intuitive proof for the 180 degree triangle. Simply imagine walking around the perimeter of the triangle, starting and stopping with the same orientation. That means that you turned through a total of 360 degrees while circumnavigating the triangle, which means the 3 exterior angles of the triangle sum to 360 degrees. If the interior angles of the triangle are A, B, C, then that means that 180 - A + 180 - B + 180 - C = 360, which simplifies to A + B + C = 180
2:33: The sum of all of the angles of a pentagon is (5-2)*180=540, making the average angle measure to be 540/5=108, with the lines connecting the triangle angles and the pentagon angles making them supplementary angles. Supplementary angles add up to 180, and subtracting the average of 108 from 180, the average angle measure of the non-lettered angle to be 72. 72*2(amount of non-lettered angles)=144, so the average of the lettered angles to be 36. 36*5=180, so there we go!
3:15 All the triangles you need are already there. The inner angle of the pentagon, say opposing a, forms a triangle with b and e. Hence the corresponding outer angles of the pentagon are b+e. Repeat with outer angles opposing e, obtain a+d. Now these two outer angles form a triangle with c. We conclude a+b+c+d+e = 180° #
This idea applies to integrals to me. I marvel at the things you know to do. It's unlike derivatives, which have formulas. Similarly, solving anything with W(x) seems hard, too. Or proving derivative rules from the limit definition of a derivative.
When dealing with these sorts of polygons, I always imagine myself traversing along the lines and turning at each vertex. In the case of the triangle, I will have rotated exactly 360 degrees as I traverse. In the case of your star, I would have to rotate a total of 720 degrees (there will be two times that as I rotate around a vertex, I'll be facing 'up' for a moment and I end up facing the same direction I started). Then, I note how many straight-lines I travel and 'turn off'. Triangle is 3, star is 5. So, the sum of interior angles is *180 degrees minus For triangle, 3*180 - 360 = 180. For 'star' you get 5*180 - 720 = 180. For square you get 4*180 - 360 = 360. Hexagon 6*180 - 360 = 720 and so on. Only tricky bit is noting how many times you 'rotate' around as you traverse the diagram.
My frustrations due to the fact that I relate to this on an ungodly level is higher than the value of any number divided/0 Quite non existent, but hypothetically a lot
What I thought was the following: You can divide the pentagon into tree triangles, thus any pentagon has 540 degrees. Each corner of the pentagon shares the line of where an angle of a triangle lies. The value of that angle in that triangel is 180 - the value of that naoboring angle in the pentagon. Because the whole pentagon is 540 deg, the average pentagon angle is 108 deg. 180-108 makes that the average triangel angle that is naoboring to the pentagon is 72 deg. Double that because each triangel contains two such angles and get 144 deg. 180 - 144 deg to get the average angle of the angles we are interested in. That is 36, and time it with 5 such angles to get 180 deg.
Oh man, I'm taking the Geometry Fundamental course on Brilliant, and boi, even tho there is an in-depth guidance to the set of skills I want to acquire, it is still hard to look at possible clues every time a new daily challenge on geometry pops out. Yes, you are right. It is difficult, and I need a lot of practice to gain experience. :D
Personally geometry is the easier part of Maths for me , probably for my engineer studies, algebra sometimes could be really heavy to resolve without any tiny fail
The "ok, I understand how it was solved, but how on Earth do you come up with the solution in the first place?" is not exclusive of geometry. All math is full of that. Cryptography and internet security is based on math problems for which a solution is very hard to find but very easy to verify. It is called the P=NP conjecture. Now, these 2 are not particularly good examples. While these 2 are more practical ways to solve it (or to show the truth of the statements), they are not the most intuitive ones. I myself figured it out for any polygon that self-intersects or not. Let me show how it works in these 2 examples. Triangle: walk from a to b, turn on b towards c, turn on c towards a, and turn on a to end up facing in the same direction that you started. You turned a total of 360 degrees. But how much of these 360 degrees you turned at each corner? Well, if you go from a to b and then turn back towards a, you turned 180 degrees (1/2 circle), but you turn b degrees less than that, so you turned 180-b. It is the same in each of the 3 corners, so you turned 360=(180-a)+(180-b)+(180-c)=3*180-(a+b+c), so the sum of the internal angles = a+b+c = 3*180-360 = 180 What if you have a "normal" quadrilateral? You then have 4 corners to turn so it is 4*180-360=360 What about a "normal" pentagon? 5*180-360=540 What about that special pentagon called pentagram (also known as "star")? Well you still have 5 sides so you still have 5*180, but when you walk around the star you complete not one but 2 full turns, so it is 5*180-720. Yup, that's 180. See? No special knowledge of geometry needed. Just to know that each full turn is 180 degrees. No "magic trick" required to be able to solve it, just walk around the polygon.
I love geometry the most because it requires not only skill but also intuition. Sometimes it is very frustrating to solve a hard geometry problem, but if one thinks with clarity and uses his intuition then he must figure it out. So, sir I think geometry is more interesting than calculas. But calculas is not easy. Every topic of math is difficult unless mastered.
Same! Even I'm not good at doing it, geometry is one of the most beautiful creatures in mathematics. Geometry is more about imagination than calculating! The Theory Of General Relativity, the current best description of gravity is also a theory of geometry.
@@Mysoi123, I'll never say I'm good at geometry. But I solve most of the geometry problems myself. I wanna give you a precious advice. Please take it well. Try to solve any geometry problem on your own. Don't ask your teacher to show you how to solve.(unless it is totally out of your reach). You will automatically do good. God bless you.
What your opinion on co-ordinate geometry ..like dealing with circles , parabola , hyperbola , ellipse using the co-ordinates..Give some videos regarding this
The star one can proven even more easily using the property of exterior angle example Take the triangle containing d the angle on the right hand side will be equal to a+c and other b+e then in the triangle d+(a+c)+(b+e)=180
Another takeaway is to make the star circumscribed by a circle. That is, each angle (a - e) makes an intercepted arc that is twice of the corresponding angle which then adds up to 360. Since all inscribed angles are half of the intercepted arcs, then the sum of angles is 360/2 = 180.
I typically view the courses Arithmetic, Algebra, Geometry and Trigonometry as the "foundation" for math, in that specific course order - in my opinion, math courses start to reach high school level when you get to Differential Calculus and Linear Algebra, and then university level maybe at courses like Multivariable Calculus and Partial Differential Equations. Seems about right to me.
I highly recommend you find a proof of the derivative of sin x that does not use the angle-sum formulae, especially if you think geometry is too hard. I actually did this twice without finding the indeterminate value of (sin x)/x.
On the star problem, I have a different fun way to solve it. (Well, fun to me!) Let point A be where angle a is, etc. Now extend the two lines that make a down past the star and do the same for the two lines that make e. Now make the line that is parallel to line BD but goes through the previous four lines down below the star. To me the diagram now looks like the star is floating above the ground and shining searchlights on it out of points A and E. :) Fun! The line DA intersects the parallel at (let's call it) D'. The line BE intersects the parallel at (let's call it) B'. The line CA intersects the parallel at (let's call it) C'. The line CE intersects the parallel at (let's call it) C". Triangle AD'C' has interior angle a at A (opposite angles) and interior angle d at D' (alternate interior angles - Thanks, 5th Postulate!). That makes the *exterior* angle at C' equal to their sum, a+d. Triangle EB'C" has angle e at E (opposite angles) and angle b at B' (alternate interior angles - Thanks, 5th Postulate!). That makes the *exterior* angle at C" equal to their sum, b+e. But that means triangle CC'C" (which, as a triangle, has 180⁰) has angle sum c+(a+d)+(b+e)=a+b+c+d+e=180⁰, as required. :-D
What u feel for geometry is the same thing I feel for integration jee adv questions,how in the world are we going to know what Trick or manipulation should be used???
Simple. Just ask yourself what if it(the question) was something different which I've solve before. That makes things a lot easier. And if you are really talking about JEE integrals then you have to remember few(about 20 or so) standard forms.
The whole reason geometry is "hard" is because you need to develop mathematical creativity and intuition. It is just plain intuition to draw that parallel line or perpendicular line. Many people who work with geometry often try and fail so many times at first and think so hard, but after years, they get more experienced. The best way to learn geometry is to discover and let your mind wander and find patterns in shapes. Try to discover theorems and prove theorems and just appreciate the beauty of geometry. Geometry is simply hard creative work, and you have to think more creatively. I also struggle in some Geometry Problems(especially olympiad and competition math) , but I keep trying and thinking hard, which is why I love geometry. Geometry makes you think.
It depends on the person. Terrence Tao said that algebra and topology are his weaknesses and that he only gets a good understanding of them by translating the problem to analysis and geometry.
in algebra, for example when you first encounter: factor 2xx+3x+1, you don't just simply get the idea to split 3x in 2x+x Also, there's lots of things in geometry that we can do without much creativity, if you know how to construct something with a ruler and compass, you can using trigonometry and analytical geometry methods, easiliy find the required lengths, or areas, because coordinates of line and circle intersections can be easily calculated
Great video! I would love to see more geometry problems but done with a calculus approach. I just came across the "quadrature of a parabola" from Archimedes which says that the area of a parabolic segment is 4/3 of the inscribed triangle. I've been trying to figure out how to prove it with calculus...
Y'all need Descartes ;) His _La Geometrie_ shows how to bounce back and forth between drawings and quadratic equations when doing geometry and unlocks quite a lot for someone coming from an analysis background Here's a fun one: draw a regular heptagon and use it to find sin(2 pi/7), similar to how you can draw a pentagon to find sin(2 pi/5). It can be done but you need the cubic formula...
Geometry problems are always the ones who scare me the most. They require a lot of imagination. Chinese math culture focuses a lot on these kind of problems since young age, I wonder if it helps with creating a good mathematical mindset.
Hi, Here's a question A graph has an equation y=√x^2-9 ( the x^2 and the -9 are inside the sqrt ). It is restricted from x=3 to x=5. The region is bounded by the y=0 line and those 2 x values. If the region was rotated about the y axis, what would be the correct resulting solid of revolution for it? My answer was π ∫y^2+9 dy, from 4 to 0, but the 'correct' answer is π ∫16-y^2 dy, from 4 to 0
Strictly speaking, in the first problem that parallel line exists without using the 5th postulate. But it's the 5th postulate that lets us say that the two blank angles on either side of b (let's call them a' and c') actually equal a and c, respectively. Even without the 5th postulate, a'+b+c' = 180⁰, but without the 5th postulate, a+b+c could still be strictly less than 180⁰ without breaking the other 4 postulates. (a+b+c can't be greater than 180⁰ without breaking some other of the 4 postulates - can't remember which one right now - which is why Hyperbolic Geometry counts as an Absolute Geometry, but Elliptical/Riemannian Geometry is not an Absolute Geometry.)
@@neu3478 hey i am jee aspirant and these questions are really simple for us Thus why wherever you see jee aspirant comments that's mean it basically simple question for us.... Try jee advanced questions that refresh u .. 😂
Are you following Michael Penn's attempt to get better in geometry? He started to prove theorems in his channel (Thales, Pythagoras). And a bonus, my favorite joke: a right triangle with sides marked as "1", "i" and hypotenuse as "0". P.S. That's not quite a joke. That's the structure of Minkowski's 4-space.
I mainly wanted to answer why I think geometry is hard. I should have also talked about why I don't do that many geometry videos (bc the drawing is very hard, for example th-cam.com/video/U3Isvh5ffuY/w-d-xo.html)! Michael Penn is great in all aspects! And to answer you question "do I also want to get better in geometry?" Maybe one day, lol.
@@black_jack_meghav Einstein was his student, actually :) And yes, they both worked on relativity. Actually there is a plenty of people who worked on relativity, don't think Einstein did all that himself.
Geometry is a important part of mathematics. And it is the main thing in our life. We need it and use it for many important works like measuring Land .
Several months ago on programmers forum I saw problem which was very hard to solve in terms of pure symbolic math (formulas). System of cubic equatations and more arises. But some smart ass distinguished it's geometric nature and reformulate it as system of vectors and circles with defined lengths being in touch in specific points. And problem was gone. It became as simple as cross product with some simple vector additions. So I think symbolic math and geometry could be in touch and must be. ;)
When integrating using substitution, how do you know what substitution to use? That seems as difficult to me. Edit: I am not seeking advice on how to integrate, I am just pointing out that it is just as tricky as geometry. And so are many other areas of maths. You need to get the right idea. What is easy depends o what you have experience with.
Ok, I hope this can help We know that taking an indefinite integral (an integral with no known bounds) can prompt us to take the antiderivative of the function. Thus, the expression inside the integral can be seen as something that has been differentiated to give the expression. Now, the rationale behind u substitution I find (and in a textbook I borrow) is that that expression inside has undergone the use chain rule. Thus, applying the correct u substitution can allow us to get the integral down into a non-composite function ( opposed to a function like f[g(x)] ) in-terms of another variable. If you have a really keen eye (and/or have been using the chain rule to death) you can recognize the expression inside as a result of the chain rule and not even have to consider u-substitution (I am not sure in-terms of validity in an exam situation, but it is a valid technique and it has been called the reverse chain rule). Consider taking the antiderivative of (2xsin(x^2)) By substituting u=x^2, we differentiate to give du/dx=2x, which is equivalent to du=2xdx thus, sin(x^2)2xdx----> sin(u)du Taking the antiderivative of sin(u) is -cos(u), thus the antiderivative of 2xsin(x^2)=-cos(x^2)+c Now, either by noticing the chain in the example, or by differentiating the term on the RHS, we could see we could have straight up anti differentiated the expression in one step.
Maybe another example is (cos(x))^3*-sin(x). as we see, if we let u=cos(x), then du/dx=-sin(x), and thus du=-sin(x)dx and thus we can continue anti differentiating from there. Also, if my explanation is not the best, I am sorry since I am not a teacher, but a student that will most likely start integration with my class after the 2 week break I am currently having. Good luck to you in your studies.
@@wenhanzhou5826 the thing about the recognize step is probably that you do enough chain rule to recognize the chain rule in the intergrand, ie looking for structure as a result of application of chain rule. Here, I feel like I can get an x^2 u substitution because the derivative of x^2 appears as a product with the sine term.
wow. its cool but difficult. These proofs are very cool and i guess it would take a long time to observe spot them, since ive little experience and I guess im close minded when it comes to trying different approaches.
Another proof is idk how to write it in math but.. take a line that is 180°.. then take your hand and fold it to make the ends touch each other.. there u make a triangle that is of 180°
Onctly i figure out by my self the proof of the star : simply join any two vertices , consider the opposite triangle form by the opposite equal angles and name the two angle form by the new triangle( x , y ) you get x+y = a+c (wlog ) .....
Here is an interesting way for both proofs Proof 1 Imagine a small stick with an arrow along AC, now rotate the stick about C by angle C so that its now along BC, Next rotate it about B by angle B so now its along BA NOW Rotate about A by angle A, so now its along CA. So since Net rotation of stick is 180 degrees A +B+ C=180 Proof 2 Start stick with an arrow along AD, rotate by D so, along BD Rotate about B, now its along BE, Rotate about E so now its along CE Rotate about C so now its along CA Rotate qbout A so now its along DA Since net rotation 180 degrees A+B+C+D+E=180
If you bake a cake in a square pan at 90 degrees, the corners will come out right.
Lmao what a joke 😆😆😆😆😆😆😄😊😆
Well what if I bake a square cake at 0 degrees 🤔?
My pan is π/2.
@@gavasiarobinssson5108 yes.
Radians is way better
Should be in a quadrilateral tin at 90degrees
Everyday, he looks more like a wizard. My man is evolving.
Lmfao you're actually right
u mean "math sorceror" ?
He needs a staff.
@@MathAdam He needs a pointy hat as well.
@@pronounjow Then he can stand at the door and greet students before exams: "YOU SHALL NOT PASS!!!"
Summary: lack of practice!
Clever man. While protesting how hard geometry is, he carries us through 2 beautiful proofs.
I teach my students that two of the most powerful tricks in Algebra are adding zero and multiplying by 1. When you complete the square you add zero, and when you add fractions you are multiplying each fraction by 1. In a proof class you have to do this more often and more creatively.
It's similar with geometry. You have to create. But this time instead of creating a clever 1 or 0, you are creating a line or other geometric entity that expands your diagram. It's just a different form of creativity. It comes naturally to some people and less naturally to others.
One of the reasons why geometry is one of my favourite areas of Mathematics is because of the thought that it requires. You have to look at things outside of the box (no pun intended) and you are rewarded for your creativity. Due to this, there are an endless number of possible solutions to just one geometry question.
Great post. Geometry, in my opinion, is the most important subject in high-school. Best class to develop pure conceptual problem solving. Applies to all parts of education/life.
@@heatmyzer9 It saddens me when people say stuff like "I'm not even gonna need to do maths in my future career so there's no point of even paying attention." Maths is so much more than knowing how to crunch numbers, it also MASSIVELY develops critical thinking and creativity.
@@staticchimera44 Agree 100%. Life/success involves solving all kinds of problems and making decisions. If you can’t do those two things….your path through life is greatly narrowed. 0.01% direct use of geometry by most….99% indirect application through creative problem solving.
@@staticchimera44 agree. The math problems might not be useful in your daily life but knowing how to solve problems gets way better by solving such problems
@@heatmyzer9 one of the best but physics is more important in our life
Speaking about the difficulty of synthetic geometry, I would love you to make a video about using complex numbers in geo problems. I'm sure a lot of people don't know about that topic.
It's very tedious to work out though
:)
Fact : Calculus is easier than geometry
I am an asian kid (13) I approve It's so hard aaaaaaaaaaa pain....aaa
@@vishalmalviya89 As another asian kid (14) I partly disagree.
@@segmentsAndCurves Triangles questions aren't hard but 2-3 figures merged together get me scared up..
@@vishalmalviya89 Yeah, that really scary.
What about a discrete set of points?
@@segmentsAndCurves gave up on that
Thanks for saying this! I watch quite a few geometry videos on TH-cam, and I'm always asking myself how you're supposed to know where to start. Most videos are like, "now do this random arbitrary thing that just lets you solve the problem," as if it's obvious what to do.
Try to prove the “9-point circle” if you want a geometry challenge.
Here’s how to draw the 9-point circle th-cam.com/video/EJWUI4s3-tU/w-d-xo.html
Just take a homothety at H with ratio 1/2
My friend gave this particular qn to me 2-3 yrs ago. And I was able to solve it in 1 period(40 mins) in school but got into trouble getting the notes what the teacjer had done during the class.
Yes its easy just prove a bunch of cyclic quadrilaterals
@@rudradate3691 I did it by exterior angles of a triangle. That is sum of 2 opp interior angles = exterior angle.
@@rikthecuber oh ok .... but where did you use this property?
Finally, someone who thinks geometry is hard too...and yes, exactly, you gotta be very observative to know how to do the problems...when I was at 7th grade I loved geometry because it was quite easy...at 8th grade I started to hate it a little bit because it was harder, it required more observation to do it...same thing at 9th grade, I hated it even more, now I finished 10th grade and what i can say, I love algebra so much...and I already watched some of your videos with calculus, which I should study in my next 2 years, and I learned a lot from those videos, thanks to you
I completely understand what you say. I have lived the same and then later I came to love those challenges. And that is why I can and should advice you to learn to not hate what challenges you, but take the challenge as an oportunity to grow and improve. Otherwise nothing of value can be achieved. THE KEY TO MAKE IT is to start where you are not strugling a lot but just a little, then from there progress with a pace that is still comfortable but not far from too fast with the right guidance of a good teacher, tutor or coach. Have fun, make friends in those batles, do it for the fun or exploring and the thrill of discovering, for the joy of intelectual growth. Enjoy!
Didficult geometry problems always take way more time to solve than equally difficult algebra/precalc problems. I mean I remember we had some olympid level problems given to us to solve at home and I ended up just sitting and looking at the god damn drawing for hours.
@@nektarsolne4niy804 that's me too 😂
This is exactly what i went through
Yeah I absolutely hate geometry, specially proofs
I agree with you sir from my experience as an IMO participant.
Solutions that depend only on Euclidean geometry are really nice but really tough to discover.
How in the world can we know that we need to construct this line, draw this circle, or even define new points?!
Luckily there are other tools that made my life easier (Trigonometry - complex coordinates - barycentric coordinates - projective geometry).
I have discussed these useful tools in short in one of my videos about what topics one should study to prepare for math Olympiad contest.
I also have just started a geometry tutorial on my channel!
So yes Geo is hard, but that's exactly what makes it beautiful 😉
Yes I agree with you. Some of them are very nice but some are very complex.
well said!!!
wow... can you do some videos on probability? I love this.
And also combinatorics. I really need to know how many derangements of "Mississippi" and "aabbcc."
@36243 MARCO DANIEL GOMES DErangements, not arrangements.
Being very used to algebra really make reasoning in geometric terms really hard, at least for me. That's why reading through Euclid's Element is harder for me than reading through calculus books.
What blew my mind was that the ancient greek used geometric reasoning for everything, that's how we get the terms like square of a number, completing the squre and so on. Don't even get me started on Appolonius' Conics, it literally melted my brain.
I'd say that the argument "how in the world do I know to do it" isn't specific to geometry. Just review your own algebra/calculus videos to see you're doing tricks nobody without experience will come up with. Looking at your "my first quintic" or "a brilliant limit"
No, in algebra, you don’t even need the actual slick choices (even numerical methods and computational)
You just need the option of having a slick choice
I agree but he stated that he doesn’t have much experience in geometry, and that was a primary reason for it being more difficult.
Jordan Curve theorem
Poincare conjecture
are impossible to prove
@@duckymomo7935 That is true. Algebra you can use brute force, take a long as time but it can be done. For example, expanding a binominal. You can either know the coefficients and powers, or you can literally sit there multiplying out. It's ugly as hell, but doable.
@@Tony29103 Disagree. It's not all about powers and binomials in algebra. There are problems which will make even experienced people stuck. Examples - equations like these:
√(x-1/𝑥)+√(1-1/𝑥)=𝑥
∛(𝑥+22)=𝑥³+6𝑥²+12𝑥-32
𝑥+𝑥/√(𝑥²-1)=35/12
etc; for such you'll need to know a trick or sometimes a combination of right methods (and still stumble for a while) - and good luck "brute-forcing" those with raising powers, that's lead you nowhere close to solving them.
I like to prove the star one thinking about the external angles of the pentagon and the 5 triangles. Because actually, the angles a,b,c,d and e are the only ones that aren't from the external angles of the pentagon. So we have that the sum of the 15 angles from the triangles must be 900⁰, but the 10 angles from the bases of the triangles are actually 2 times the external angles of the pentagon, so they are 2×360⁰=720⁰, then we have 720⁰+a+b+c+d+e=900⁰, a+b+c+d+e=180⁰
To be honest the "star" problem isn't hard... There's a simple solution using exterior angle of triangle where you will obtain a triangle with the angles equal to b, a+d, c+e hence a+b+c+d+e=180 degrees.
There is also a more intuitive proof for the 180 degree triangle. Simply imagine walking around the perimeter of the triangle, starting and stopping with the same orientation. That means that you turned through a total of 360 degrees while circumnavigating the triangle, which means the 3 exterior angles of the triangle sum to 360 degrees. If the interior angles of the triangle are A, B, C, then that means that 180 - A + 180 - B + 180 - C = 360, which simplifies to A + B + C = 180
@@XJWill1 that's a really nice proof.
I have been waiting for this kind of geo with bprp and i love it .
2:33: The sum of all of the angles of a pentagon is (5-2)*180=540, making the average angle measure to be 540/5=108, with the lines connecting the triangle angles and the pentagon angles making them supplementary angles. Supplementary angles add up to 180, and subtracting the average of 108 from 180, the average angle measure of the non-lettered angle to be 72. 72*2(amount of non-lettered angles)=144, so the average of the lettered angles to be 36. 36*5=180, so there we go!
3:15 All the triangles you need are already there. The inner angle of the pentagon, say opposing a, forms a triangle with b and e. Hence the corresponding outer angles of the pentagon are b+e. Repeat with outer angles opposing e, obtain a+d. Now these two outer angles form a triangle with c. We conclude a+b+c+d+e = 180° #
Its beauty lies in the imagination.
4:28 Blue pen!!! 😲😁😊
Great video!
This idea applies to integrals to me. I marvel at the things you know to do. It's unlike derivatives, which have formulas.
Similarly, solving anything with W(x) seems hard, too. Or proving derivative rules from the limit definition of a derivative.
When dealing with these sorts of polygons, I always imagine myself traversing along the lines and turning at each vertex. In the case of the triangle, I will have rotated exactly 360 degrees as I traverse. In the case of your star, I would have to rotate a total of 720 degrees (there will be two times that as I rotate around a vertex, I'll be facing 'up' for a moment and I end up facing the same direction I started).
Then, I note how many straight-lines I travel and 'turn off'. Triangle is 3, star is 5. So, the sum of interior angles is *180 degrees minus For triangle, 3*180 - 360 = 180. For 'star' you get 5*180 - 720 = 180. For square you get 4*180 - 360 = 360. Hexagon 6*180 - 360 = 720 and so on. Only tricky bit is noting how many times you 'rotate' around as you traverse the diagram.
Hey I just commented the same thing! This is exactly how I was thinking too - it's so much simpler. This comment deserves way more likes!!!
And this is just 2D Euclidean geometry. Sometimes you need parallel hyperplanes. Sometimes, you don't even have the tool of parallel lines!
Me : Oh no this strategy failed, I should try to approach the problem from another angle
The problem : Haha draw the figure again
My frustrations due to the fact that I relate to this on an ungodly level is higher than the value of any number divided/0
Quite non existent, but hypothetically a lot
What I thought was the following:
You can divide the pentagon into tree triangles, thus any pentagon has 540 degrees.
Each corner of the pentagon shares the line of where an angle of a triangle lies. The value of that angle in that triangel is 180 - the value of that naoboring angle in the pentagon.
Because the whole pentagon is 540 deg, the average pentagon angle is 108 deg. 180-108 makes that the average triangel angle that is naoboring to the pentagon is 72 deg.
Double that because each triangel contains two such angles and get 144 deg. 180 - 144 deg to get the average angle of the angles we are interested in.
That is 36, and time it with 5 such angles to get 180 deg.
Oh man, I'm taking the Geometry Fundamental course on Brilliant, and boi, even tho there is an in-depth guidance to the set of skills I want to acquire, it is still hard to look at possible clues every time a new daily challenge on geometry pops out. Yes, you are right. It is difficult, and I need a lot of practice to gain experience. :D
Personally geometry is the easier part of Maths for me , probably for my engineer studies, algebra sometimes could be really heavy to resolve without any tiny fail
You can also calculate the angle sum with exterior angle (that’s the first thing I learn with exterior angle) :)
When I did my Geometry/Trig, I also used the unit circle with line and angles to help understand.
The "ok, I understand how it was solved, but how on Earth do you come up with the solution in the first place?" is not exclusive of geometry. All math is full of that. Cryptography and internet security is based on math problems for which a solution is very hard to find but very easy to verify. It is called the P=NP conjecture.
Now, these 2 are not particularly good examples. While these 2 are more practical ways to solve it (or to show the truth of the statements), they are not the most intuitive ones. I myself figured it out for any polygon that self-intersects or not. Let me show how it works in these 2 examples.
Triangle: walk from a to b, turn on b towards c, turn on c towards a, and turn on a to end up facing in the same direction that you started. You turned a total of 360 degrees. But how much of these 360 degrees you turned at each corner? Well, if you go from a to b and then turn back towards a, you turned 180 degrees (1/2 circle), but you turn b degrees less than that, so you turned 180-b. It is the same in each of the 3 corners, so you turned 360=(180-a)+(180-b)+(180-c)=3*180-(a+b+c), so the sum of the internal angles = a+b+c = 3*180-360 = 180
What if you have a "normal" quadrilateral? You then have 4 corners to turn so it is 4*180-360=360
What about a "normal" pentagon? 5*180-360=540
What about that special pentagon called pentagram (also known as "star")? Well you still have 5 sides so you still have 5*180, but when you walk around the star you complete not one but 2 full turns, so it is 5*180-720. Yup, that's 180.
See? No special knowledge of geometry needed. Just to know that each full turn is 180 degrees. No "magic trick" required to be able to solve it, just walk around the polygon.
I love geometry the most because it requires not only skill but also intuition. Sometimes it is very frustrating to solve a hard geometry problem, but if one thinks with clarity and uses his intuition then he must figure it out. So, sir I think geometry is more interesting than calculas. But calculas is not easy. Every topic of math is difficult unless mastered.
Same!
Even I'm not good at doing it, geometry is one of the most beautiful creatures in mathematics.
Geometry is more about imagination than calculating!
The Theory Of General Relativity, the current best description of gravity is also a theory of geometry.
@@Mysoi123, I'll never say I'm good at geometry. But I solve most of the geometry problems myself. I wanna give you a precious advice. Please take it well. Try to solve any geometry problem on your own. Don't ask your teacher to show you how to solve.(unless it is totally out of your reach). You will automatically do good. God bless you.
@@Mathematician6124 Thank you!
What your opinion on co-ordinate geometry ..like dealing with circles , parabola , hyperbola , ellipse using the co-ordinates..Give some videos regarding this
Heuristics applied in solving problem. Equally applicable to algebra.
The star one can proven even more easily using the property of exterior angle example
Take the triangle containing d the angle on the right hand side will be equal to a+c and other b+e then in the triangle d+(a+c)+(b+e)=180
Right... If BPRP thinks geometry is hard, I'm in good company. I feel liberated.
to improve your skills in geometry you should learn (examine) a lot of different geometrical constructions
I love this video on the star - wonderful to see the proof
Another takeaway is to make the star circumscribed by a circle. That is, each angle (a - e) makes an intercepted arc that is twice of the corresponding angle which then adds up to 360. Since all inscribed angles are half of the intercepted arcs, then the sum of angles is 360/2 = 180.
I typically view the courses Arithmetic, Algebra, Geometry and Trigonometry as the "foundation" for math, in that specific course order - in my opinion, math courses start to reach high school level when you get to Differential Calculus and Linear Algebra, and then university level maybe at courses like Multivariable Calculus and Partial Differential Equations. Seems about right to me.
I highly recommend you find a proof of the derivative of sin x that does not use the angle-sum formulae, especially if you think geometry is too hard. I actually did this twice without finding the indeterminate value of (sin x)/x.
On the star problem, I have a different fun way to solve it. (Well, fun to me!) Let point A be where angle a is, etc. Now extend the two lines that make a down past the star and do the same for the two lines that make e. Now make the line that is parallel to line BD but goes through the previous four lines down below the star.
To me the diagram now looks like the star is floating above the ground and shining searchlights on it out of points A and E. :) Fun!
The line DA intersects the parallel at (let's call it) D'.
The line BE intersects the parallel at (let's call it) B'.
The line CA intersects the parallel at (let's call it) C'.
The line CE intersects the parallel at (let's call it) C".
Triangle AD'C' has interior angle a at A (opposite angles) and interior angle d at D' (alternate interior angles - Thanks, 5th Postulate!). That makes the *exterior* angle at C' equal to their sum, a+d.
Triangle EB'C" has angle e at E (opposite angles) and angle b at B' (alternate interior angles - Thanks, 5th Postulate!). That makes the *exterior* angle at C" equal to their sum, b+e.
But that means triangle CC'C" (which, as a triangle, has 180⁰) has angle sum c+(a+d)+(b+e)=a+b+c+d+e=180⁰, as required.
:-D
Glad you liked it! ☺️
What u feel for geometry is the same thing I feel for integration jee adv questions,how in the world are we going to know what Trick or manipulation should be used???
Simple. Just ask yourself what if it(the question) was something different which I've solve before. That makes things a lot easier. And if you are really talking about JEE integrals then you have to remember few(about 20 or so) standard forms.
The whole reason geometry is "hard" is because you need to develop mathematical creativity and intuition. It is just plain intuition to draw that parallel line or perpendicular line. Many people who work with geometry often try and fail so many times at first and think so hard, but after years, they get more experienced. The best way to learn geometry is to discover and let your mind wander and find patterns in shapes. Try to discover theorems and prove theorems and just appreciate the beauty of geometry. Geometry is simply hard creative work, and you have to think more creatively. I also struggle in some Geometry Problems(especially olympiad and competition math) , but I keep trying and thinking hard, which is why I love geometry. Geometry makes you think.
The values for tan(67.5°) and tan(22.5°) can be obtained with an octagon.
It depends on the person. Terrence Tao said that algebra and topology are his weaknesses and that he only gets a good understanding of them by translating the problem to analysis and geometry.
great vid dude! Thanks
in algebra, for example when you first encounter: factor 2xx+3x+1, you don't just simply get the idea to split 3x in 2x+x
Also, there's lots of things in geometry that we can do without much creativity, if you know how to construct something with a ruler and compass, you can using trigonometry and analytical geometry methods, easiliy find the required lengths, or areas, because coordinates of line and circle intersections can be easily calculated
Great video! I would love to see more geometry problems but done with a calculus approach. I just came across the "quadrature of a parabola" from Archimedes which says that the area of a parabolic segment is 4/3 of the inscribed triangle. I've been trying to figure out how to prove it with calculus...
Y'all need Descartes ;) His _La Geometrie_ shows how to bounce back and forth between drawings and quadratic equations when doing geometry and unlocks quite a lot for someone coming from an analysis background
Here's a fun one: draw a regular heptagon and use it to find sin(2 pi/7), similar to how you can draw a pentagon to find sin(2 pi/5). It can be done but you need the cubic formula...
Geometry questions: *comes up*
Last 3 brain cells: Adios
"You just have to be really, observative"
Like thats ever gonna happen
Curious, can you do analytic geometry? Thats geometry on a coordinate plane.
Geometry problems are always the ones who scare me the most. They require a lot of imagination. Chinese math culture focuses a lot on these kind of problems since young age, I wonder if it helps with creating a good mathematical mindset.
Hi,
Here's a question
A graph has an equation y=√x^2-9 ( the x^2 and the -9 are inside the sqrt ). It is restricted from x=3 to x=5. The region is bounded by the y=0 line and those 2 x values. If the region was rotated about the y axis, what would be the correct resulting solid of revolution for it?
My answer was π ∫y^2+9 dy, from 4 to 0, but the 'correct' answer is π ∫16-y^2 dy, from 4 to 0
I wish you would do more geometry!
Strictly speaking, in the first problem that parallel line exists without using the 5th postulate. But it's the 5th postulate that lets us say that the two blank angles on either side of b (let's call them a' and c') actually equal a and c, respectively. Even without the 5th postulate, a'+b+c' = 180⁰, but without the 5th postulate, a+b+c could still be strictly less than 180⁰ without breaking the other 4 postulates. (a+b+c can't be greater than 180⁰ without breaking some other of the 4 postulates - can't remember which one right now - which is why Hyperbolic Geometry counts as an Absolute Geometry, but Elliptical/Riemannian Geometry is not an Absolute Geometry.)
Indian jee aspirants : this is our childhood favourite questions 😂
Yes 😂😂
Stoppppp the jeee comments AAAAAA
@@neu3478 yes those comments are cringe
@@Usuario459 Less than you
@@neu3478 hey i am jee aspirant and these questions are really simple for us
Thus why wherever you see jee aspirant comments that's mean it basically simple question for us....
Try jee advanced questions that refresh u ..
😂
Are you following Michael Penn's attempt to get better in geometry? He started to prove theorems in his channel (Thales, Pythagoras).
And a bonus, my favorite joke: a right triangle with sides marked as "1", "i" and hypotenuse as "0".
P.S. That's not quite a joke. That's the structure of Minkowski's 4-space.
Did minkowski work with Einstein on relativity?
I mainly wanted to answer why I think geometry is hard. I should have also talked about why I don't do that many geometry videos (bc the drawing is very hard, for example th-cam.com/video/U3Isvh5ffuY/w-d-xo.html)! Michael Penn is great in all aspects!
And to answer you question "do I also want to get better in geometry?"
Maybe one day, lol.
@@black_jack_meghav Einstein was his student, actually :)
And yes, they both worked on relativity. Actually there is a plenty of people who worked on relativity, don't think Einstein did all that himself.
@@nikitakipriyanov7260 ahh thanks that's great 😃
Geometry is a important part of mathematics. And it is the main thing in our life. We need it and use it for many important works like measuring Land .
Several months ago on programmers forum I saw problem which was very hard to solve in terms of pure symbolic math (formulas). System of cubic equatations and more arises. But some smart ass distinguished it's geometric nature and reformulate it as system of vectors and circles with defined lengths being in touch in specific points. And problem was gone. It became as simple as cross product with some simple vector additions. So I think symbolic math and geometry could be in touch and must be. ;)
This is so cool!! I love it.😊😊
I would love if you add more GEO We have enough calculus and series please more GEOMETRIE
I always considered geometry as one of the purest ideas of mathematics.
It is purely problem solving and proofs
I request a series where you try to solve geometry questions
for the triangle one you can also use the boomerang method to find the sum of 3 angles then use the paralel line thing again
Ah yes my favourite method, *Boomerang method*
@@78anurag we call it like that where I come from
@@yigityagz6599 ok
I love geometry, the proofs are so cool
If you look at the problems of IMO, you can notice that geometry problems mostly easier than other problems
Once for all A MATH youtuber that speaks FACTS
I like geometry. There’s so many ways to answer on question. Problem solving is fun.
When integrating using substitution, how do you know what substitution to use? That seems as difficult to me.
Edit: I am not seeking advice on how to integrate, I am just pointing out that it is just as tricky as geometry. And so are many other areas of maths. You need to get the right idea. What is easy depends o what you have experience with.
I think most people use 'u'
Ok, I hope this can help
We know that taking an indefinite integral (an integral with no known bounds) can prompt us to take the antiderivative of the function. Thus, the expression inside the integral can be seen as something that has been differentiated to give the expression. Now, the rationale behind u substitution I find (and in a textbook I borrow) is that that expression inside has undergone the use chain rule. Thus, applying the correct u substitution can allow us to get the integral down into a non-composite function ( opposed to a function like f[g(x)] ) in-terms of another variable. If you have a really keen eye (and/or have been using the chain rule to death) you can recognize the expression inside as a result of the chain rule and not even have to consider u-substitution (I am not sure in-terms of validity in an exam situation, but it is a valid technique and it has been called the reverse chain rule).
Consider taking the antiderivative of (2xsin(x^2))
By substituting u=x^2, we differentiate to give du/dx=2x, which is equivalent to du=2xdx
thus, sin(x^2)2xdx----> sin(u)du
Taking the antiderivative of sin(u) is -cos(u), thus the antiderivative of 2xsin(x^2)=-cos(x^2)+c
Now, either by noticing the chain in the example, or by differentiating the term on the RHS, we could see we could have straight up anti differentiated the expression in one step.
Maybe another example is (cos(x))^3*-sin(x). as we see, if we let u=cos(x), then du/dx=-sin(x), and thus du=-sin(x)dx and thus we can continue anti differentiating from there.
Also, if my explanation is not the best, I am sorry since I am not a teacher, but a student that will most likely start integration with my class after the 2 week break I am currently having. Good luck to you in your studies.
@@justinpark939 thing is how do you do the "recognize" step?
@@wenhanzhou5826 the thing about the recognize step is probably that you do enough chain rule to recognize the chain rule in the intergrand, ie looking for structure as a result of application of chain rule. Here, I feel like I can get an x^2 u substitution because the derivative of x^2 appears as a product with the sine term.
I can't tell you how many times drawing a smiley face saved me in geometery. 😁
Is there a proof about the angles of a triangle that can be done in non-Euclidean geometry?
wow. its cool but difficult. These proofs are very cool and i guess it would take a long time to observe spot them, since ive little experience and I guess im close minded when it comes to trying different approaches.
Thanks sir i like it so much
*gasp* the fabled blue pen!
Please do the Integration e^(x+1/x)
I do geometry. It breathes art in my work.
It also is the foundation for the flower/seed of life. Why not make a mathematical principle around it?
"Observative", I like it!
Another proof is idk how to write it in math but.. take a line that is 180°.. then take your hand and fold it to make the ends touch each other.. there u make a triangle that is of 180°
I will make a channel in the end of 2021
So subscribe now if you want (if not then do not subscribe)
Onctly i figure out by my self the proof of the star : simply join any two vertices , consider the opposite triangle form by the opposite equal angles and name the two angle form by the new triangle( x , y ) you get x+y = a+c (wlog ) .....
When you try to solve a geometry problem using a synthetic technique your mind 😨
When you try to solve a geometry problem using bashing your mind 😎
no, they are both the top face
That thumbnail tho, almost spat out my tea
I love math and did really well in honors algebra 1 last year in 8th grade but I’m taking honors geometry and it’s so annoying
Here is an interesting way for both proofs
Proof 1
Imagine a small stick with an arrow along AC, now rotate the stick about C by angle C so that its now along BC,
Next rotate it about B by angle B so now its along BA
NOW Rotate about A by angle A, so now its along CA.
So since Net rotation of stick is 180 degrees A +B+ C=180
Proof 2
Start stick with an arrow along AD, rotate by D so, along BD
Rotate about B, now its along BE,
Rotate about E so now its along CE
Rotate about C so now its along CA
Rotate qbout A so now its along DA
Since net rotation 180 degrees
A+B+C+D+E=180
What's even funnier that we always tend to look at visual proofs for other fields like algebra, calc, and visual proofs are basically geometry :D
Where did you get the Star problem?.....I have seen it before but I can't remember
Synthetic geometry is a topic that maths enthusiasts return to when they get older and want to broaden their abilities.
keep telling yourself that.
@@centralprocessingunit4988 sorry ik it's been 2 years but: ?
can you explain more about limits
Wowq clever!
yeah i sometimes think geometry is hard
i admit:
algebra > geometry
easy peasy >? oh wow sometimes so hard
Intuition
Idk i've found geometry to be more accessible personally
geometry was the one math class i was actually excited to learn and i geniunely thought it was fun
I remember that your birthday is on the 18th so HAPPY BIRTHDAY!
Thank you BPRP
Geometry was my favorite thing in math.
I still remember I had to solve more harder geometric theorem proofs than these in my 8th grade back in the school
Finally someone who understand my pain
0:25 hey dude, the only way i know is to use creativity and the knowledge of the theorems, if you fon't have those so geometry is hard
James Tanton has a great series on proving angle sum theorems intuitively using „pencil pushing“: th-cam.com/video/gE4tin6QkY0/w-d-xo.html