Circle Through Three Points

From CGAFaq

Let the three given points be a, b, c. Use a_x and a_y to represent the x and y coordinates of a, and so on. The coordinates of the center p=(p_x,p_y) of the circle determined by a, b, and c are:

A = b_x - a_x

B = b_y - a_y

C = c_x - a_x

D = c_y - a_y

E = A*(a_x + b_x) + B*(a_y + b_y)

F = C*(a_x + c_x) + D*(a_y + c_y)

G = 2*(A*(c_y - b_y)-B*(c_x - b_x))

p_x = (D*E - B*F) / G

p_y = (A*F - C*E) / G

If G is zero then the three points are collinear and no finite-radius circle through them exists. Otherwise, the radius of the circle is:

r² = (a_x - p_x)² + (a_y - p_y)²

Reference:

• [O’Rourke (C)] p. 201. Simplified by Jim Ward.