What is the formula for finding the coordinates of circumcentre?

Answer 1

The coordinates of the circumcenter of a triangle can be found using the intersection of the perpendicular bisectors of the triangle's sides. If the vertices of the triangle are given as ( A(x_1, y_1) ), ( B(x_2, y_2) ), and ( C(x_3, y_3) ), the circumcenter's coordinates ( (D, E) ) can be calculated using the following formulas:

[ D = \frac{(x_1^2 + y_1^2)(y_2 - y_3) + (x_2^2 + y_2^2)(y_3 - y_1) + (x_3^2 + y_3^2)(y_1 - y_2)}{(x_1)(y_2 - y_3) + (x_2)(y_3 - y_1) + (x_3)(y_1 - y_2)} ]

[ E = \frac{(x_1^2 + y_1^2)(x_3 - x_2) + (x_2^2 + y_2^2)(x_1 - x_3) + (x_3^2 + y_3^2)(x_2 - x_1)}{(y_1)(x_3 - x_2) + (y_2)(x_1 - x_3) + (y_3)(x_2 - x_1)} ]

These coordinates ( (D, E) ) represent the circumcenter of the triangle.