Baltic Way 2017 · Problem 12

Let $T$ be the second intersection point of $P Q$ and $S_{1}$ and $R$ be the second intersection point of $P Q$ and $S_{2}$. Let $\angle P X T=\alpha$, $\angle R Y Q=\beta$. It is evident that $P X=X T, R Y=Y Q$. Calculate the ratio area ${ }_{P X T} /$ area $_{R Y Q}$ by two different ways. First,

$$\frac{\operatorname{area}_{P X T}}{\operatorname{area}_{R Y Q}}=\frac{X P^{2} \sin \alpha}{Y Q^{2} \sin \beta}$$

Second,

Equating these expressions we obtain

$$\frac{\operatorname{area}_{P X T}}{\operatorname{area}_{R Y Q}}=\frac{P T}{R Q}=\frac{2 R_{1} \sin \alpha}{2 R_{2} \sin \beta}$$ $$\frac{X P}{Y Q}=\sqrt{\frac{R_{1}}{R_{2}}}$$