Why every inscribed angle is the same

2026-05-12

Pin two points on a circle. Call them A and B. Now stand on the circle, anywhere on the far side, and look at A and B. The angle between them, as you see it, is a number.

Walk around the circle. The view changes - A swings one way, B swings the other. You'd expect the angle to change with the view. It does not. From every point on the far arc, you see A and B open up the exact same way.

Drag the blue dot around the major arc. The two blue rays swing about wildly. The angle between them, written inside the wedge, does not budge.

The 2:1 trick

Look at the angle at the centre - the warm wedge between the two radii. It's bigger than the inscribed angle. Twice as big. Exactly twice.

central angle  =  2 · inscribed angle

That's the inscribed angle theorem, and it's why every view from the far arc gives the same number: they're all half of the same central angle, and the central angle isn't moving.

Why 2:1, in one picture

Slide P until it sits directly opposite the chord, on the line from the centre out through the far arc. Now draw the line from P straight through the centre to the other side. You've split the inscribed angle in two, and split the central angle in two, with the same line.

Each half is now a triangle with two equal sides - the radius from the centre to A, and the radius from the centre to P. Two equal sides means two equal base angles. The exterior angle of that triangle (at the centre) equals the sum of the two opposite interior angles - both of which are the inscribed angle's half.

One half of the central angle equals two halves of the inscribed angle. Double it: the central angle is two inscribed angles. That's the whole proof.

A free corollary

Drag A and B until they sit on opposite ends of a diameter. The chord is now a straight line through the centre. The central angle is 180°. The inscribed angle is half of that.

any angle in a semicircle is exactly 90°

Thales of Miletus is supposed to have sacrificed an ox over this fact, two and a half thousand years ago. He had not yet seen the more general theorem the ox was a corner of. We just did.

Two points on a circle make a chord. From every other point on the far arc, the chord subtends the same angle. The angle at the centre is twice it. The semicircle - 90° - is the special case.