Try this Drag any orange dot. The angle ∠QRP will always be a right angle.
Put another way: If a triangle has, as one side, the diameter of a circle, and the third
vertex of the triangle is any
point on the circumference of the circle,
then the triangle will always be a right triangle.
In the figure above, no matter how you move the points P,Q and R, the triangle PQR is always a
right triangle, and the angle
∠PRQ is always a right angle.
A practical application - finding the center of a circle
The converse of Thales Theorem is useful when you are trying to find the center of a circle.
In the figure on the right, a
is on the circle always "cuts off" a diameter of the circle. That is,
the points P and Q are always the ends of a
Since the diameter passes through the center, by drawing two such diameters the center is found at the point where the diameters
For an animated demonstration of this technique see Find the Center of a Circle with a Right-angled Object.
Other circle topics
Equations of a circle
Angles in a circle
(C) 2009 Copyright Math Open Reference. All rights reserved