A circle with center $O$ and a point $A$ in this circle are given. Let $P_{B}$ is the intersection point of $[AB]$ and the internal bisector of $\angle AOB$ where $B$ is a point on the circle such that $B$ doesn't lie on the line $OA$, Find the locus of $P_{B}$ as $B$ varies.
Point $A$ is inside circle $(O)$, not on it.
Perpendicular bisector of $OP_B$ cuts $OA$ at $Q$.
$QO = QP_B = \frac{OP_B}{2 \cos \frac{_1}{^2} \angle BOA} = \frac{OA \cdot OB}{OA+OB} = \text{const}$
$\Longrightarrow$ $P_B$ is on circle $(Q)$ with radius $QO$.
