Proof of tangent segment theorem.

Statement: The segments of the tangent drawn from a point outside (external point) to a circle are congruent.

Proof: 

Construction: Draw two segments AP and AQ.

diagram of tangent segment theorem

According to the figure, A is the centre of the circle. The tangents drawn through point D from outside the circle touches the circle at the points P and Q. 

In ∆PAD and ∆QAD,  

seg PA ≅ [segQA] [Radii of the same circle]  

seg AD ≅ seg AD [Common side] 

∠APD = ∠AQD = 90° [Tangent theorem]  

∴ ∆PAD = ∆QAD [By Hypotenuse side test]  

∴ seg DP = seg DQ [congruent sides of congruent triangles]

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