A circle theorem: an angle with its vertex on a circle and sides formed by two chords intercepts an arc whose measure is twice the measure of the inscribed angle; equivalently, an inscribed angle equals half the central angle subtending the same arc.

Principle

Angle–arc correspondence: inscribed angles that subtend the same arc are equal, and every inscribed angle measures half the measure of its intercepted arc (or of the central angle for that arc), reflecting the circle's symmetry about its center.

Demonstration

On circle with center O, take points A,B,C on the circumference with vertex at B forming ∠ABC; compare ∠ABC to central angle ∠AOC subtending arc AC: compute via isosceles triangles OAB and OBC to find ∠ABC = 1/2 ∠AOC, hence half the arc measure.

Misapplication

Applying the half-arc rule to an angle whose vertex lies off the circle (for example an exterior or interior noninscribed angle) or confusing arcs with oriented measures can lead to wrong conclusions; also misusing when dealing with directed arcs in nonstandard metric contexts.

Consequence

Gives a quick method to compare angles subtending the same arc, to prove equality of angles in cyclic quadrilaterals, to locate tangent–chord angle relations, and to reduce many circle-angle problems to arc measure computations.

Reversal

Contrast with central angles: a central angle subtending an arc measures twice any inscribed angle that subtends that same arc; reversing the relation moves from boundary-based angle to center-based angle and changes vertex locus from circle to center.

Boundary

Valid in Euclidean circle geometry and in any context where arc measures and inscribed/central angles are defined; it does not hold for arbitrary curved loci or when angles are measured in non-Euclidean metrics without reinterpretation.

Semantic Tension

Can be conflated with related theorems (e.g., tangent–chord theorem) because both relate arcs and angles; the tension is between vertex-on-circle (inscribed) and vertex-at-center (central) formulations and between directed and unsigned arc measures.

Synthesis

The inscribed angle theorem ties a boundary angle to the arc it intercepts: any angle with vertex on a circle equals half the central angle (or arc measure) that subtends the same chord, enabling many angular equalities in cyclic configurations.