The inscribed angle theorem is one of the first really surprising theorems you run into in geometry class in high school or junior high school. I remember thinking, "What? Really?" when Mrs. Morgan showed it to us. Here is a diagram that illustrates the theorem and shows something else that is really remarkable and useful about the inscribed angles.

To restate the theorem: If you have any two points on a circle, such as X and Y which are at the ends of the red dashed line, the angle they form from any other point on the circle, such as P, R and even Q which is way out in left field, is the same. In this diagram, they are called angle A -- all equal.

The proof is not hard if you know just a little geometry, and there is a fairly clear

**proof on Wikipedia** that only uses basic ideas. Here is an animated drawing shown there:

One way to think about this is that the distance between those points and the size of the circle determine all those angles completely -- no matter which third point you pick on the circle, you get the same angle. Of course if you make the circle larger with the same two points, the angle will get smaller and vice versa.

The second amazing thing I mentioned above is that if the third point you choose is the center of the circle, as in the drawing above marked C, then the angle the center sees out to the starting points X and Y (called the central angle), is exactly twice the angle A, which is the inscribed angle. This central angle is usually much easier to figure out than the inscribed angle.

Here's a simple question to see if you have followed all of this: What is the central angle between the balls two apart as in Brian's demonstration? Hint: all 15 balls of the rack are uniformly spaced around the circle.