Prove that : The tangent segments drawn from an external point to the circle are congruent.[Figure: a circle with centre O and an external point P; tangent segments PA and PB touch the circle at A and B; seg OP, seg OA and seg OB are drawn.]

Proof:

Given: A circle with centre O and an external point P; PA and PB are tangent segments, touching the circle at A and B.

To prove: PA = PB.

Construction: Draw seg OP, seg OA and seg OB.

Proof: OA ⊥ PA and OB ⊥ PB (a tangent is perpendicular to the radius at the point of contact).

∴ Δ OAP and Δ OBP are right-angled triangles with right angles at A and B.

In Δ OAP and Δ OBP:
OA = OB .......... (radii of the same circle)
OP = OP ........... (common hypotenuse)

∴ Δ OAP ≅ Δ OBP .......... (hypotenuse-side test)

∴ PA = PB .......... (corresponding sides of congruent triangles)

Hence the tangent segments drawn from an external point to a circle are congruent.