Question

The perpendicular bisectors of two sides of a triangle meet at point that belongs to the third side. Prove that this is a right triangle.

Answer 1

Step-by-step explanation:

The point of intersection of the perpendicular bisectors of a triangle is the circumcenter, the center of the circle that contains the three triangle vertices.

If that center is on one side, that side must be a diameter of the circle. The diameter cuts the circle into two arcs, each of which measures 180°.

The third vertex of the triangle and its two legs form an inscribed angle that subtends an arc of 180°. The measure of that angle is half the measure of the arc, so the angle measures 90° and is a right angle.

A triangle with a right angle is a right triangle. QED