GeoProof — Intersecting Chords Angle explorer

Four points A, B, C, D sit on a circle in order. Two chords are drawn as the diagonals AC and BD, and they cross at an interior point P. Drag any of the four points around the circle. The theorem: the angle at P between the chords equals half the sum of the two arcs it cuts off — angle APB equals one half of arc AB plus arc CD. The two intercepted arcs are highlighted, and the measured angle and the half-sum of the arcs are shown live and stay equal. An info button opens a drawer with the explanation.

∠APB = ½(arc AB + arc CD)

Intersecting Chords Angle

When two chords cross inside a circle, the angle they make is the average of the two arcs they cut off:

∠APB = ½(arc AB + arc CD)

The angle "looks at" two arcs — the one in front of it (arc AB) and the one behind it through the vertical angle (arc CD) — and lands exactly halfway between their measures.

Why ½ the sum

∠APB is an exterior angle of triangle PBC, so it equals ∠PBC + ∠PCB. Those are inscribed angles standing on arcs AB and CD, so each is half its arc — and the sum is ½(arc AB + arc CD).

Compare the cousins

Vertex on the circle (inscribed angle): ½ of one arc. Vertex inside (here): ½ the sum. Vertex outside (two secants): ½ the difference. The vertex's position slides the formula from sum to difference.