Two circles intersect at points A and B. Point X lies on line AB. Prove that the lengths of the tangents drawn from point X to the circles are equal.

Question

Two circles intersect at points A and B. Point X lies on line AB but not on segment AB. Prove that the lengths of all the tangents drawn from point X to the circles are equal.

Let the tangents intersect the circles at $P_1$ and $P_2$. We have $\angle XP_1A = \angle P_1BA$ and $\angle XP_2A = \angle P_2BA$. I am not able to proceed. Please help.

Thanks.

Answer 1

Hint:

By the tangent-secant theorem we have: $$ \overline{XP_1}^2=\overline{XB}\cdot \overline{XA}=\overline{XP_2}^2 $$