Circle Tangent problem

Question

Triangle $ABC$ is isosceles with $AB=AC$. A circle that is tangent to line $AB$ at $B$ intersects line $AC$ at points $P$ and $Q$. Prove that $BC$ bisects angle $ \widehat{PBQ}$.

Answer 1

Since $AB$ is a tangent to the circle we have $\angle ABP=\angle BQP$. So $\angle QBC=\angle ACB-\angle BQP=\angle ABC-\angle BQP=\angle ABC-\angle ABP=\angle PBC$. So $BC$ bisects $\angle ABC$.