There is a notion of directional angles which are defined as the angle you need to rotate line l1 counter clockwise (of clockwise if you like) to get line l2, notive then that the angle is no longer symmetric, instead we have angle(l1,l2)=-angle(l2,l1) angle(l1,l2)+angle(l2,l3)+angle(l3,l1)=0 (you can add 180 if it makes you feel better but rotating a line 180 degrees does not change its direction) and then you don't have to check for all cases, just make sure that you have all the directions right, so for the first theorem it will look like (I will use a as short for angle): a(AB,AC)=a(AB,AO)+a(AO,AC)=x+y a(BO,CO)=a(BO,AO)+a(AO,CO)=a(BO,AB)+a(AB,BO)+a(AO,AC)+a(AC,CO)=x+x+y+y=2a(AB,AC) And this works regardless of where a is in the circle, it will just mean that some of the angles might be nagtive but the underlying computation is unchanged.
I see what you are saying - because the proofs here show that the x-y and x+y cases are essentially the same - we're using all the same angles and lines. There's an abstract step needed to talk about 'negative' angles, but if we can get over that then the proofs will all be simplified. I suppose we can think of it in two ways - either this is a neat way of summing up what the video shows, or we can start again. There's a really interesting point about formalism and proof here. I haven't fully thought about it, but in a way what you're suggesting is (or could be extended to) saying we could take the circles out of the circle theorems! In that we'd need a precise definition of angles in the abstract sense you've given, and some rules about how to combine them. The statements of the and proofs of these theorems would probably then be quite simple. The major down-side of this is you would have to do a lot more work to convince people that would have anything to do with circles, so - but on the plus side you might also find some other contexts which have the same structure and formalism - that the proofs about circles might also be proofs about something entirely different as well - then we're getting into some seriously interesting pure maths. Anyway, I think the thing I was trying to say is that as both a mathematician and a teacher, I think what I consider the 'best' proof depends a lot on who I'm talking to. In a pure maths, speaking to mathematicians who understand this sort of thing, the abstract one is elegant and effective. Speaking to students, the pictures of circles makes more sense - but that's also the fun of going further in maths I suppose, seeing how these seemingly different areas end up having the same structure. Thanks for a thought-provoking comment - I don't usually give these sorts of replies - will add this to my list of things to think about making some more content around one day!
Very nice! What are you animating in?
Thanks - I actually worked with a professional animator who did quite a few days work on this video - they use Illustrator and other Adobe software.
There is a notion of directional angles which are defined as the angle you need to rotate line l1 counter clockwise (of clockwise if you like) to get line l2, notive then that the angle is no longer symmetric, instead we have
angle(l1,l2)=-angle(l2,l1)
angle(l1,l2)+angle(l2,l3)+angle(l3,l1)=0
(you can add 180 if it makes you feel better but rotating a line 180 degrees does not change its direction)
and then you don't have to check for all cases, just make sure that you have all the directions right, so for the first theorem it will look like (I will use a as short for angle):
a(AB,AC)=a(AB,AO)+a(AO,AC)=x+y
a(BO,CO)=a(BO,AO)+a(AO,CO)=a(BO,AB)+a(AB,BO)+a(AO,AC)+a(AC,CO)=x+x+y+y=2a(AB,AC)
And this works regardless of where a is in the circle, it will just mean that some of the angles might be nagtive but the underlying computation is unchanged.
I see what you are saying - because the proofs here show that the x-y and x+y cases are essentially the same - we're using all the same angles and lines. There's an abstract step needed to talk about 'negative' angles, but if we can get over that then the proofs will all be simplified. I suppose we can think of it in two ways - either this is a neat way of summing up what the video shows, or we can start again.
There's a really interesting point about formalism and proof here. I haven't fully thought about it, but in a way what you're suggesting is (or could be extended to) saying we could take the circles out of the circle theorems! In that we'd need a precise definition of angles in the abstract sense you've given, and some rules about how to combine them. The statements of the and proofs of these theorems would probably then be quite simple.
The major down-side of this is you would have to do a lot more work to convince people that would have anything to do with circles, so - but on the plus side you might also find some other contexts which have the same structure and formalism - that the proofs about circles might also be proofs about something entirely different as well - then we're getting into some seriously interesting pure maths.
Anyway, I think the thing I was trying to say is that as both a mathematician and a teacher, I think what I consider the 'best' proof depends a lot on who I'm talking to. In a pure maths, speaking to mathematicians who understand this sort of thing, the abstract one is elegant and effective. Speaking to students, the pictures of circles makes more sense - but that's also the fun of going further in maths I suppose, seeing how these seemingly different areas end up having the same structure.
Thanks for a thought-provoking comment - I don't usually give these sorts of replies - will add this to my list of things to think about making some more content around one day!
The best. Thanks for all.
Thank you!
Wish I discovered your channel earlier, actual W of a video
thanks!