Two right triangles PRQ and PSQ are drawn on the same hypotenuse PQ . If PR and QS intersect at T , prove that $\displaystyle \mathrm{ST} \times \mathrm{TQ}=\mathrm{PT} \times \mathrm{TR}$.

$\displaystyle \begin{array}{l}\Delta \mathrm{STP} \sim \Delta \mathrm{RTQ} \frac{\mathrm{ST}}{\mathrm{TP}}=\frac{\mathrm{RT}}{\mathrm{TQ}} \Rightarrow \mathrm{ST} \times \mathrm{TQ}=\mathrm{PT} \times \mathrm{TR}\end{array}$