The vertices of a triangle lie on the same circle, and one of its sides is the diameter. Prove that this triangle is right-angled.

The vertices of the triangle lie on the same circle. One side is its diameter. Let us prove that the triangle is right-angled.

To begin with, note that if the vertices of a triangle lie on a circle, then the circle is circumscribed around the triangle.

One of the sides of the triangle is its diameter – we are talking about the hypotenuse of a right-angled triangle. The center of the circle is located in the middle of the hypotenuse of the triangle, respectively, the hypotenuse is twice as long as the radius of the circle.