GeoProof — Tangent–Chord Angle explorer

tangent–chord angle = inscribed angle = ½ × arc TB

Drag A around its arc — the inscribed angle stays equal to the tangent–chord angle. Cross A to the other arc and both become the supplementary pair.

The Tangent–Chord Angle

Draw a tangent to a circle at a point T, and a chord TB from the point of tangency. The angle between the tangent and the chord equals the inscribed angle that the same chord TB subtends from the opposite arc:

∠(tangent, TB) = ∠TAB

This is often called the alternate segment theorem: the tangent–chord angle equals any inscribed angle in the alternate segment of the circle.

Both equal half the arc

Each of these angles is exactly half the arc TB it cuts off. That is why they match — and why the inscribed angle stays constant as you slide A along its arc, just like in the inscribed-angle theorem.

Why the tangent gives half the arc

Think of the inscribed angle from a point that slides all the way up to T itself: in the limit, the chord to that point becomes the tangent line. So the tangent direction behaves like "the chord from T to T," and the angle it makes with TB is still half the arc.

The two angles at T

The tangent line has two sides, giving two tangent–chord angles that add up to 180°. Each one equals the inscribed angle in its own alternate segment. Drag A across the chord to the other arc and watch which angle it matches.

A bit of history

It appears in Euclid's Elements, Book III, Proposition 32 (c. 300 BCE), as part of the classical theory of circles and tangents.