an inscribed angle

Now stand not at the centre but on the rim of the circle, and again open your arms to two other rim points. The angle you make there is an inscribed angle. Remarkably, if you slide your viewpoint to any other spot on the same arc, the angle you see those two points at does not change — every seat on that arc views the chord at the same angle.

Precisely, an inscribed angle is an angle whose vertex lies on the circle and whose two sides are chords reaching to two other points on the circle. The inscribed angle theorem is its headline property: an inscribed angle is exactly half the central angle subtending the same arc — equivalently, half the measure of the arc it intercepts. So if an arc measures 100 degrees, every inscribed angle standing on it measures 50 degrees, wherever its vertex sits on the rest of the circle. Two immediate corollaries follow: inscribed angles subtending the same arc are equal (the 'same-segment' property), and an inscribed angle in a semicircle — standing on a diameter, a 180-degree arc — is a right angle (this last is Thales' theorem).

The inscribed angle theorem is the workhorse of circle geometry, behind cyclic quadrilaterals, the tangent-chord angle, and countless angle-chases. The proof splits into cases by where the centre falls relative to the angle, but the cleanest case (one side a diameter) uses an isosceles triangle of two radii and the exterior-angle theorem to show the central angle is twice the inscribed one. The standing misconception: do not assume any angle inside a circle is half its arc. The halving holds only when the vertex is ON the circle. A vertex at the centre gives the full arc; a vertex strictly inside or outside gives a different rule again (averages and half-differences of two arcs).