parmenides51 wrote: Let $ABC$ be an acute-angled triangle and let $P$ be the intersection of the tangents at $B$ and $C$ of the circumscribed circle of $\vartriangle ABC$. The line through $A$ perpendicular on $AB$ and cuts the line perpendicular on $AC$ through $C$ at $X$. The line through $A$ perpendicular on $AC$ cuts the line perpendicular on $AB$ through $B$ at $Y$. Show that $AP \perp XY$. Let $O$ be the circumcenter of $\vartriangle ABC$.Let $AP \cap (OBPC) = D.$ We just need to prove that $O,D,X,Y$ are collinear. $\angle ODP = \angle OBP =90^\circ$ $\angle BAY = \angle CAX = A$ $\angle AYB = \angle AXC =90^\circ -A$ $\angle BCP = \angle BDP = 90^\circ - A = \angle AYB$ So, $D$ lies on $(ABY).$ Similarly , $D$ lies on $(AXC).$ $\angle ADY = \angle ABY = 90^\circ$ $\angle ODP =\angle ADY = 90^\circ$ and $A,D,P$ are collinear. This means $O,D,Y$ are collinear. Similarly $O,D,X$ are also collinear

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Solution. WLOG, $AC\geq AB$. Clearly, $AXA'Y$ is a parallelogram and $BY$ and $CX$ concur at the antipode of $A$ on $(ABC)$, say $A'$. Define $L$ and $M$ to be the midpoints of $\overline{BC}$ and $\overline{AA'}$. Since $AP$ is a symmedian of $\bigtriangleup ABC$, it suffices to show that $\angle BYX=\angle LAC$. However, this is true, because $AL$ and $YM$ are corresponding cevians of the similar triangles $\bigtriangleup BAC$ and $\bigtriangleup AYA'$. $\blacksquare$

Let AP meet circumcircle of BCP at Z. we will prove Z is where those lines are perpendicular. ∠BZP = ∠BCP = ∠BAC = ∠BYA so BZAY is cyclic. ∠CZP = ∠CBP = ∠CAB = ∠CXA so CZAX is cyclic. ∠BZY = ∠BAY and ∠CZX = ∠CAY. ∠BZY + ∠CZB + ∠CZX = ∠BAY + 2∠BAC + ∠CAY = ∠CAY + ∠BAX = 180. so Y,Z,X are collinear.

This problem is an application of the properties of isometrics. It is easy to prove that if the line passing through the point X and perpendicular to the line AX intersects the line passing through the point Y and perpendicular to the line AY at the point T, then AT is the median line of the triangle ABC. And AP is the center line of the accompany position, the proposition is established.