Solves circumcenter of a triangle using \[C_{\triangle}=\left[\frac{x_1\cdot\sin2A+x_2\cdot\sin2B+x_3\cdot\sin2C}{\sin2A+\sin2B+\sin2C}\right]\]
The circumcenter of a triangle is the point where the perpendicular bisectors of the sides of the triangle intersect. In other words, it is the center of the circle that passes through all three vertices of the triangle.
To find the circumcenter of a triangle, follow these steps:
Draw the triangle and label its vertices as A, B, and C.
Find the midpoint of each side of the triangle.
The midpoint of side AB is M, the midpoint of side BC is N, and the midpoint of side AC is P.
Construct the perpendicular bisectors of each side.
To do this, draw a line perpendicular to side AB that passes through point M. Label the intersection of this line with side AB as D.
Similarly, draw a line perpendicular to side BC that passes through point N. Label the intersection of this line with side BC as E.
Lastly, draw a line perpendicular to side AC that passes through point P. Label the intersection of this line with side AC as F.
The point where the three perpendicular bisectors intersect is the circumcenter of the triangle. Label this point as O.
Note: If the three perpendicular bisectors do not intersect at a single point, then the triangle is not a valid triangle.
