Central Angle Theorem

The central angle theorem states:

The central angle drawn from any two points on a circle is twice as large as any inscribed angle drawn from those two points.

Explanation and proof:

Draw a circle, and label three points on it $A$, $B$ and $P$. Also label the center of the circle $O$.

The points can be anywhere you want as long as $P$ is not on the same side as $A$ and $B$.

Now construct the following line segments: $\overline{OA}$, $\overline{OB}$, $\overline{PA}$, and $\overline{PB}$.

The angle $\angle AOB$ is an example of a central angle, and the angle $\angle APB$ is an example of an inscribed angle.

Now construct the line segment $\overline{OP}$ and note that all radii have the same length.

Since $\triangle AOP$ and $\triangle ABP$ have two equal line segments, they are isosceles and their base angle must be congruent. Label the congruent angles $x$ and $y$.

The interior angles of a triangle must sum to $180^\circ$, so angles $\angle AOP$ and $\angle BOP$ must measure $180-2x$ and $180 -2y$, respectively.

The angles around the origin must add up to $360^\circ$. Therefore,

Subtracting 360 from both sides and rearranging, we have

Meanwhile, we can see from the last diagram that

Therefore, we must conclude that $\angle AOB$ has twice the measure of $\angle APB$.

Q.E.D.

Note: this theorem only applies if $P$ lies on the longer of the two arcs connecting $A$ and $B$. If $P$ lies on the minor arc, then the equation becomes

This is why I insisted earlier that $P$ be on the side opposite to $A$ and $B$.