Inscribed Angle Theorem

Inscribed angle = half the central angle on the same arc.

The idea

Theorem (Inscribed angle theorem).

Let $A$ and $B$ be two points on a circle with center $O$, and let $C$ be a third point of the circle. Then the inscribed angle $\angle ACB$ is half the central angle $\angle AOB$ subtending the same arc $AB$, the arc not containing $C$: $\angle ACB = \tfrac12\,\angle AOB.$

The theorem relates two angles that subtend the same arc of a circle. An angle subtends an arc when the arc lies between its two sides. With $A$ and $B$ the endpoints of the arc, the central angle $\angle AOB$ has its vertex at the center $O$ and its sides along the radii $OA$ and $OB$, and the inscribed angle $\angle ACB$ has its vertex at the point $C$ on the circle and its sides $CA$ and $CB$.

The theorem never mentions where the vertex $C$ sits, so moving $C$ along its arc changes nothing: every inscribed angle subtending a given arc has the same measure, half the central angle.

Ways to work on it

• Walkthrough. Convert between central and inscribed angles.
• Proof. See why it's half — the isosceles-triangle picture.
• Practice. Random central/inscribed angle conversions.
• Hardest. Thales' theorem and the semicircle right angle.

Not sure where to start? Take the ten-question placement test.