Thales' Theorem

The diameter of a circle always subtends a right angle to any point on the circle
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 above, a right angle whose vertex 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 diameter line.

Since the diameter passes through the center, by drawing two such diameters the center is found at the point where the diameters intersect.

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