Angles from Tangents & Secants

Every circle angle rule depends on one thing: where the vertex is.

• At the centre: the angle equals the arc.
• On the circle: half the arc.
• Inside (two chords crossing): half the *sum* of the two intercepted arcs.
• Outside (two secants, two tangents, or one of each): half the *difference* of the two intercepted arcs.

That is the whole family, and it is far easier than memorising six separate theorems.

Notice the pattern: as the vertex moves from the centre outward, the angle shrinks. Full arc, then half, then half the sum, then half the difference.

For the outside case, always subtract the near arc from the far one. A negative answer means you did it backwards.

The four cases are unified by vertex position relative to the circle.

Centre: θ = arc. On the circle: θ = ½·arc. Interior intersection of two chords: θ = ½(arc₁ + arc₂), where the arcs are the two intercepted by the vertical pair. Exterior: θ = ½|arc_far − arc_near|.

The interior and exterior formulas both follow from the Inscribed Angle Theorem plus the Exterior Angle Theorem applied to an auxiliary triangle formed by joining two endpoints of the chords or secants — each is a sum or difference of two inscribed angles.

The exterior case covers three configurations (two secants, two tangents, or a tangent and a secant) with the same formula, since a tangent is the limiting case of a secant whose two intersection points coincide, making its two "endpoints" identical.

For two tangents from an external point, the two arcs sum to 360°, so the angle simplifies to 180° − arc_near, giving the useful fact that the angle between two tangents and the near arc are supplementary.