2013 USAMO 第 1 题

In triangle $ABC$, points $P,Q,R$ lie on sides $BC,CA,AB$ respectively. Let $\omega_A$, $\omega_B$, $\omega_C$ denote the circumcircles of triangles $AQR$, $BRP$, $CPQ$, respectively. Given the fact that segment $AP$ intersects $\omega_A$, $\omega_B$, $\omega_C$ again at $X,Y,Z$ respectively, prove that $YX/XZ=BP/PC$.

在三角形$ABC$中，点$P、Q、R$分别位于边$BC、CA、AB$上。设$\omega_A$、$\omega_B$、$\omega_C$分别表示三角形$AQR$、$BRP$、$CPQ$的外接圆。鉴于线段$AP$再次与$\omega_A$、$\omega_B$、$\omega_C$相交于$X,Y,Z$，证明$YX/XZ=BP/PC$。