Journey To The Center of a Circle

I found the below video recently, which blew my mind. Something as humble as the simple triangle, can have so many facets to it – yet again reaffirming the beauty of Maths.

One thing I found out, from watching the video, is that there are multiple kinds of centers that a triangle can have – mainly circumcenter, centroid, orthocenter and incenter.

Circumcenter

A  circumcircle of a triangle is a circle which passes through all the vertices of the polygon. The center of which, is called a circumcenter.

The Cartesian coordinates are given by

[latex size=1]((A_y^2 + A_x^2)(B_y – C_y) + (B_y^2 + B_x^2)(C_y – A_y) + (C_y^2 + C_x^2)(A_y – B_y)) / D,[/latex]

[latex size=1](A_y^2 + A_x^2)(C_x – B_x) + (B_y^2 + B_x^2)(A_x – C_x) + (C_y^2 + C_x^2)(B_x – A_x)) / D[/latex]
where
[latex size=1]D = 2( A_x(B_y – C_y) + B_x(C_y – A_y) + C_x(A_y – B_y)).[/latex]

 

Centroid

The centroid or barycenter of a triangle is the intersection of all straight lines that divide the triangle into two parts of equal moment about the line.

The Cartesian coordinates are the means of the coordinates of the three vertices.

[latex size =1]C = \frac13(a+b+c) = \left(\frac13 (x_a+x_b+x_c),\;\;
\frac13(y_a+y_b+y_c)\right)[/latex]

 

Orthocenter

The altitude of a triangle is a straight line through a vertex and perpendicular to a line containing the base (the opposite side of the triangle). When the three altitudes from the sides of the triangles intersect in a single point, it is called the orthocenter of the triangle.

Unlike the centroid and circumcenter of a triangle, the orthocenter has no special characteristics (such as being equidistant from all sides or vertices).

The triliniar coordinates are given by
[latex size=2 ]\cos{B}\cos{C}:\cos{A}\cos{C}[/latex]

Incenter

The incircle or inscribed circle of a triangle is the largest circle contained in the triangle, that touches the three sides. The center of the incircle is called the triangle’s incenter.

The Cartesian coordinates are
[latex size =2]C = \bigg(\frac{a x_a+b x_b+c x_c}{P},\frac{a y_a+b y_b+c y_c}{P}\bigg) = \frac{a(x_a,y_a)+b(x_b,y_b)+c(x_c,y_c)}{P}[/latex]

where

[latex size =2]P = a + b + c[/latex]