In triangle ABC, AB=AC.The circle through B touches the side AC at the mid-point D of AC, passes through a point P on AB. Prove that 4 ×AP = AB ?

1 Answer
Apr 19, 2017

see explanation.

Explanation:

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According to tangent-secant theorem:
"When a tangent and a secant are drawn from one single external point to a circle, square of the length of tangent segment must be equal to the product of lengths of whole secant segment and the exterior portion of secant segment."

=> AD^2=AP*AB
Given AB=AC, and D is midpoint of AC,
=> AD=(AC)/2=(AB)/2

=>((AB)/2)^2=AP*AB
=> (AB)^2/4=AP*AB
=> AB=4AP

Hence 4AP=AB