What is the length of [BC]?

Question

Let A , B and C be 3 points of a circle (c)

P is the intersection of two tangents of the circle in points B and C

Let (AB)//(CP) and AB=3 and BP=4 What is the length of BC

Can someone give hint ! Im blocked , the problem is that i don't know to start my proof

I see that BPC is triangle such that CP=BP

We suppose that ABPC is quadrilater parallelogram

Answer 1

It is not difficult to show that XBPC is a parallelogram. In fact, it is a rhombus because $PB = BX = 4.$

$\alpha = \beta$ (tangent properties)

$\beta = \gamma$ (alternate angles)

$\beta = \delta$ (angles in alternate segment)

$∴ \triangle PBC \sim \triangle CBA$

$∴ \dfrac {PB}{BC} = \dfrac {BC}{BA}$

That is, $\dfrac {4}{BC} = \dfrac {BC}{3}$

Result follows.