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Let BC be a fixed segment in the plane, and let A be a variable point in the plane not on the line BC. Distinct points X and Y are chosen on the rays (CA) ⃗ and (BA) ⃗, respectively, such that ∠CBX=∠YCB=∠BAC.Assume that the tangents to the circumcircle of ABC at B and C meet line XY at P and Q, respectively, such that the points X,P,Y, and Q are pairwise distinct and lie on the same side of BC. Let Ω1 be the circle through X and Y centred on BC. Similarly let Ω2 be the circle through Y and Q centred on BC. Prove that Ω1 and Ω2 intersect at two fixed points as A varies. 【译】在同一平面内,BC为给定线段,动点A不在直线BC上. X和Y分别为射线(CA) ⃗,射线(BA) ⃗上不重合的两点,满足∠CBX=∠YCB=∠BAC.若三角形ABC外接圆在点B和C处的切线分别交直线XY于点P和点Q,点X,P,Y,Q不重合,且位于直线BC同侧.圆Ω1经过点X,P且圆心在BC上.类地,圆Ω2经过点Y,Q且圆心在BC上.证明:当点A运动时,圆Ω1和圆Ω2始终交于两定点. |
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