Difference between Incenter, Circumcenter, Centroid & Orthocenter | Concept Clarification.

Difference between Incenter, Circumcenter, Centroid & Orthocenter | Concept Clarification | Angle Bisector | Perpendicular Bisector | Median.
In this video,we learn about Incenter, Circumcenter , Centroid & Orthocenter and Difference among them.
Incenter
In geometry, the incenter of a triangle is a triangle center, a point defined for any triangle in a way that is independent of the triangle's placement or scale. The incenter may be equivalently defined as the point where the internal angle bisectors of the triangle cross, as the point equidistant from the triangle's sides, as the junction point of the medial axis and innermost point of the grassfire transform of the triangle, and as the center point of the inscribed circle of the triangle.
Together with the centroid, circumcenter, and orthocenter, it is one of the four triangle centers known to the ancient Greeks, and the only one that does not in general lie on the Euler line. It is the first listed center, X(1), in Clark Kimberling's Encyclopedia of Triangle Centers, and the identity element of the multiplicative group of triangle centers.
Definition
The incenter lies at equal distances from the three line segments forming the sides of the triangle, and also from the three lines containing those segments. It is the only point equally distant from the line segments, but there are three more points equally distant from the lines, the excenters, which form the centers of the excircles of the given triangle. The incenter and excenters together form an orthocentric system
Circumcenter
The circumcenter of a triangle is defined as the point where the perpendicular bisectors of the sides of that particular triangle intersect. In other words, the point of concurrency of the bisector of the sides of a triangle is called the circumcenter. It is denoted by P(X, Y). The circumcenter is also the centre of the circumcircle of that triangle and it can be either inside or outside the triangle.
Circumcenter Formula
P(X, Y) = [(x1 sin 2A + x2 sin 2B + x3 sin 2C)/ (sin 2A + sin 2B + sin 2C), (y1 sin 2A + y2 sin 2B + y3 sin 2C)/ (sin 2A + sin 2B + sin 2C)]
Centroid
The centroid is the centre point of the object. The point in which the three medians of the triangle intersect is known as the centroid of a triangle. It is also defined as the point of intersection of all the three medians. The median is a line that joins the midpoint of a side and the opposite vertex of the triangle. The centroid of the triangle separates the median in the ratio of 2: 1. It can be found by taking the average of x- coordinate points and y-coordinate points of all the vertices of the triangle.
Centroid Formula
Centroid of a triangle = ((x1+x2+x3)/3, (y1+y2+y3)/3)
Orthocenter
The orthocenter is the point where all the three altitudes of the triangle cut or intersect each other. Here, the altitude is the line drawn from the vertex of the triangle and is perpendicular to the opposite side. Since the triangle has three vertices and three sides, therefore there are three altitudes.
Orthocenter of a Triangle
The orthocenter of a triangle is the point where the perpendicular drawn from the vertices to the opposite sides of the triangle intersect each other.
• For an acute angle triangle, the orthocenter lies inside the triangle.
• For the obtuse angle triangle, the orthocenter lies outside the triangle.
• For a right triangle, the orthocenter lies on the vertex of the right angle.
Orthocenter Formula
mBE = (y-y2)/(x-x2)
mAD = (y-y1)/(x-x1)
Properties of Orthocenter
The orthocenter is the intersection point of the altitudes drawn from the vertices of the triangle to the opposite sides.
• For an acute triangle, it lies inside the triangle.
• For an obtuse triangle, it lies outside of the triangle.
• For a right-angled triangle, it lies on the vertex of the right angle.
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Circumcenter: circumcenter is the point of intersection of three perpendicular bisectors of a triangle. Circumcenter is the center of the circumcircle, which is a circle passing through all three vertices of a triangle.
Incenter is the center of the circle with the circumference intersecting all three sides of the triangle.
Orthocenter is the point of intersection of the three heights of the triangle.

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The orthocentre may lie on any vertex if the triangle is right angled triangle

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