In geometry, an inscribed angle is the angle formed in the interior of a circle when two chords intersect on the circle. It can also be defined as the angle subtended at a point on the circle by two given points on the circle.

Equivalently, an inscribed angle is defined by two chords of the circle sharing an endpoint.

The inscribed angle theorem relates the measure of an inscribed angle to that of the central angle intercepting the same arc.

The inscribed angle theorem appears as Proposition 20 in Book 3 of Euclid's Elements.

Note that this theorem is not to be confused with the Angle bisector theorem, which also involves angle bisection (but of an angle of a triangle not inscribed in a circle).

The inscribed angle theorem states that an angle θ inscribed in a circle is half of the central angle 2θ that intercepts the same arc on the circle. Therefore, the angle does not change as its vertex is moved to different positions on the same arc of the circle.

Let O be the center of a circle, as in the diagram at right. Choose two points on the circle, and call them V and A. Designate point B to be diametrically opposite point V. Draw chord VB, a diameter containing point O. Draw chord VA. Angle ∠BVA is an inscribed angle that intercepts arc AB; denote it as ψ. Draw line OA. Angle ∠BOA is a central angle that also intercepts arc AB; denote it as θ.

Lines OV and OA are both radii of the circle, so they have equal lengths. Therefore, triangle △VOA is isosceles, so angle ∠BVA and angle ∠VAO are equal.