Prove that the tangents drawn at the endpoints of a chord of a circle make equal angles with the chord?

Question

Answer

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Let AB be a chord of a circle with center O and let AP and BP be the tangents at A and B.

Let the tangent meet at P.Join OP Suppose OP meets AB at C.

To prove : ∠PAC = ∠PBC

Proof: In ΔPAC and ΔPBC

PA = PB [Tangents from an external point to a circle are equal]

∠APC = ∠BPC [PA and PB are equally inclined to OP]

PC = PC [Common]

ΔPAC ≅ ΔPBC [SAS Congruence]

∠PAC = ∠PBC [C.P.C.T]

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