GeoProof — Intersecting Secants Angle explorer

angle at P = ½ · (far arc − near arc)

Drag P farther out and the angle shrinks; slide it onto the circle and the two arcs meet — the angle becomes an inscribed angle.

Intersecting Secants Angle

From a point P outside a circle, draw two secants. The first meets the circle at A (near) and B (far); the second at C (near) and D (far). The two secants cut off two arcs: a far arc BD and a near arc AC. The angle at P is half their difference:

∠P = ½ · (arc BD − arc AC)

Only the difference matters — slide both secants and as long as the two arcs change together, the angle holds.

One family, three cases

This is the "two arcs" rule for an angle whose vertex is outside the circle. When the vertex is on the circle, the near arc shrinks to zero and you get the inscribed-angle / tangent-chord rule (½ of one arc). When the vertex is inside, the rule becomes half the sum of the two arcs instead.

Secants, tangents, or a mix

The same formula covers two tangents, or a tangent and a secant, from the external point — a tangent is just the limit where a secant's two points merge, so its "arc" is measured to the single point of contact.

Why half the difference

Join one far point to the near points to make a triangle; the inscribed-angle theorem turns the two circle angles into half-arcs, and the triangle's exterior-angle rule subtracts them — leaving half the difference of the arcs at P.

A bit of history

These circle-angle relationships are part of the classical Euclidean theory of the circle (Euclid's Elements, Book III), formalized over two millennia ago.