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Revise: Circles

Tangent-radius perpendicularity, the two-tangent theorem, the tangent-chord (alternate segment) theorem, and proving four points are concyclic.

The Tangent-Radius Perpendicularity Theorem

A tangent ⊥ the radius at the point of tangency.

OT=6, OP=10 → PT=8.

The Two-Tangent Theorem

Two tangents from one external point are equal.

PA=14 → PB=14.

The Tangent-Chord Angle Theorem

Tangent-chord angle = inscribed angle in the alternate segment.

65° tangent-chord angle → 65° alternate inscribed angle.

Proving Four Points Are Concyclic

Opposite angles sum to 180° ⇒ concyclic.

∠W=110°, ∠Y=70° → concyclic.