Two acute-angled triangles ∆ABC and ∆PQR are similar. Their circumcenters are X and Y respectively. If BC and QR are corresponding (similar) sides, then prove that BX : QY = BC : QR.

Q.Two acute-angled triangles ∆ABC and ∆PQR are similar. Their circumcenters are X and Y respectively. If BC and QR are corresponding (similar) sides, then prove that BX : QY = BC : QR.

∆ABC and ∆PQR are two similar acute-angled triangles, with circumcenters X and Y respectively. If BC and QR are corresponding sides, we need to prove that BX : QY = BC : QR. Construction: Join B to X and C to X; also join Q to Y and R to Y. **Proof:** Since ∆ABC and ∆PQR are similar triangles, their corresponding sides are proportional, and so are their circumradii. Therefore, \[ \frac{AB}{PQ} = \frac{BC}{QR} = \frac{AC}{PR} = \frac{BX}{QY} \] That is, \[ \frac{BX}{QY} = \frac{BC}{QR} \] Or, BX : QY = BC : QR (Proved)