In the given fig., ABC is an isosceles triangle in which AB = AC. A circle through B touches the side AC at D and it intersects the side AB at P. If D is the mid-point of AC, prove AB = 4AP.
I used a (sort of) well-known theorem:
If a tangent and a secant are drawn to a circle from an external point,
the tangent is the mean proportional of the secant and the external segment of the secant.
is an isosceles triangle wherte
A circle through touches side at , and intersects side at .
If is the midpoint of , prove that: .Code:B ♥ o o o * * o o * * o o * * o * * o * * o o * ♥ P o* o * * * * o o * * o o * * o o * ♥ * * o ♥ o * * * * ♥ A D
We are given: .
is the midpoint of .
From the theorem, we have: . .
Substitute  into : .
. . Therefore: .