Question

Two vertices of an equilateral triangle are (−1, 0) and (1, 0) and its third vertex lies above the x-axis. Find the equation of the circumcircle of the triangle.

Answer 1

Let B = (−1, 0) and C = (1, 0) be two vertices of an equilateral triangle ABC (see Fig.). Now, BC = AC = AB = 2 and OC = OB = 1. Since the third vertex lies above the x-axis on the perpendicular  bisector of the side BC follows A = (O, k), where k = 0, we have

Therefore, A = ( 0,√3). The equation of the perpendicular bisector of AB is given by

which meets y-axis at

Therefore, centre is given by

Therefore, circumradius is given by 2√3 . Hence, the equation of the circumcircle is given by