Question 5
Let $ABC$ be an acute triangle with $AB\lt AC$, inscribed in a circle $(O)$. Let $E,F$ be the reflections of $A$ in $B,C$, respectively, and let $M$ be the midpoint of $BC$. On the line $AM$, take points $I,P$ such that triangles $AIE$ and $APF$ are isosceles at $I$ and $P$, respectively. Let $T$ be the intersection of the lines $IE$ and $PF$.
(a) Prove that $T$ belongs to $(O)$.
(b) Let $D=BI\cap CP$, $J=TD\cap IP$, and $X=OT\cap BC$. The lines $AT,AX$ meet $EF$ at $L,U$, respectively. Prove that the four circumcircles of triangles $TAJ,TIP,TEF,TLU$ have a second common point distinct from $T$.
(a) Prove that $T$ belongs to $(O)$.
(b) Let $D=BI\cap CP$, $J=TD\cap IP$, and $X=OT\cap BC$. The lines $AT,AX$ meet $EF$ at $L,U$, respectively. Prove that the four circumcircles of triangles $TAJ,TIP,TEF,TLU$ have a second common point distinct from $T$.