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.

Let ABC be an acute-angled triangle with AB > AC. Let P be the intersection of the tangents to the circumcircle of ABC at B and C. The line through the midpoints of line segments PB and PC meets lines AB and AC at X and Y respectively.

Prove that the quadrilateral AXPY is cyclic.

【译】在锐角三角形ABC中，AB>AC，△ABC的外接圆在点B和点C处的切线交于点P.一条同时过PB和PC中点的直线与AB,AC分别交于点X,Y.

求证：A,X,P,Y四点共圆.

如图所示，在△BC中，M是边AC的中点，D,E是△ABC的外接圆在点A处的切线上的两点，满足MD//AB，且A是线段DE的中点，过A,B,E三点的圆与边AC相交于另一点P，过A,D,P三点的圆与DM的延长线相交于点Q.证明：∠BCQ=∠BAC.

如图，△ABC为给定的锐角三角形，其内切圆ω分别与边AB,AC切于点K,L.高AH分别与∠ABC,∠ACB的平分线交于点P,Q.设ω₁,ω₂分别为△KPB,△LQC的外接圆，AH的中点ω₁,ω₂外，求证：从AH的中点引向ω₁,ω₂的切线相等.

如图，∠ABC=90°,AC=BC,AD⊥CE,BE⊥CE，垂足分别是D,E.已知AD=8,BE=3，则DE=______.