An* inscribed angle* is an angle formed by two chords of a circle with the vertex on its circumference. In the first circle in Figure 1, segments *AB* and *AC* are chords of a circle and the vertex *A* is on its circumference. Hence, angle *A* is an inscribed angle. In the second circle in Figure 1, angle *Q* is also an inscribed angle.

Angles formed by the two radii of a circle are called *central angles*. The vertex of a central angle is on the center of the circle. In the Figure 1 below, angles *BOC *and *SOT* are central angles.

The arc determined by the endpoints of two chords or two radii, and is opposite of the angle, is called the *intercepted arc*. In the first circle in Figure 1, arc *BC*, denoted by the red curve, is the intercepted arc of angle *BAC* (or angle *BOC*). Similarly, in the second circle, arc *ST* denoted by the green curve, is the intercepted arc of the central angle *SOT*. Saying the other way around, angle *BAC *is the *subtended* *angle* of arc *BC* and and angle *SOT* is the subtended angle of arc *ST*.

**The Inscribed Angle Theorem**

In this article, we are going to discuss the relationship between an inscribed angle and a central angle (I have created a GeoGebra applet about it) having the same intercepted arc. This is shown in the first circle in Figure 1. Angle *BAC* and angle *BOC* have the same intercepted arc *BC*. Therefore, if a central and an inscribed angle have the same intercepted arc, the measure of that central angle is twice that of the measure of the inscribed angle.

- Figure 1 – Inscribed angles, central angles and their intercepted arcs.

One strategy to solve a problem in geometry is to sometimes draw lines from one point to another. These lines drawn are called *auxiliary lines*. » Read more