(A) State and prove Basic Proportionality Theorem.

(i) \(\displaystyle \triangle \mathrm{ABC} \sim \triangle \mathrm{PQR} \Rightarrow \angle \mathrm{A}=\angle \mathrm{P}\) \(\displaystyle \frac{\mathrm{AB}}{\mathrm{PQ}}=\frac{\mathrm{AC}}{\mathrm{PR}} \Rightarrow \frac{2 \mathrm{AM}}{2 \mathrm{PN}}=\frac{\mathrm{AC}}{\mathrm{PR}}\) (as CM and RN are the medians) \(\displaystyle \therefore \triangle \mathrm{AMC} \sim \triangle \mathrm{PNR}\) (ii) \(\displaystyle \triangle \mathrm{ABC} \sim \triangle \mathrm{PQR} \Rightarrow \angle \mathrm{B}=\angle \mathrm{Q}\) \(\displaystyle \frac{\mathrm{AB}}{\mathrm{PQ}}=\frac{\mathrm{BC}}{\mathrm{QR}} \Rightarrow \frac{2 \mathrm{MB}}{2 \mathrm{NQ}}=\frac{\mathrm{BC}}{\mathrm{QR}}\) (as CM and RN are the medians) \(\displaystyle \therefore \triangle \mathrm{CMB} \sim \triangle \mathrm{RNQ}\)