ASMO Math 2022 — Grade 8 — School Round Question 22

In triangle $ABC$, the internal bisectors of $\angle A$ and $\angle B$ meet at the incenter $I$. The ray $AI$ meets the circumcircle of triangle $ABC$ again at $D$. Let $P$ be the foot of the perpendicular from $B$ to the line $AD$. Let $Q$ lie on the extension of $AD$ beyond $D$ such that $ID=DQ$.

Find the value of

$$\frac{BQ\times IB}{BP\times ID}.$$