AD and PS are medians of triangles ABC and PQR respectively such that $\displaystyle \triangle \mathrm{ABD} \sim \triangle \mathrm{PQS}$. Prove that $\displaystyle \triangle \mathrm{ABC} \sim \triangle \mathrm{PQR}$.

Since \(\displaystyle \triangle \mathrm{ABD} \sim \triangle \mathrm{PQS}\) \(\displaystyle \therefore \quad \frac{\mathrm{AB}}{\mathrm{PQ}}=\frac{\mathrm{BD}}{\mathrm{QS}}\) or \(\displaystyle \frac{\mathrm{AB}}{\mathrm{PQ}}=\frac{2 \mathrm{BD}}{2 \mathrm{QS}}\) \[\begin{aligned} & \Rightarrow \frac{\mathrm{AB}}{\mathrm{PQ}}=\frac{\mathrm{BC}}{\mathrm{QR}} \\ & \angle \mathrm{~B}=\angle \mathrm{Q} \\ & \therefore \quad \triangle \mathrm{ABC} \sim \triangle \mathrm{PQR} \end{aligned} \]