Question 6
Let $ABC$ be an acute scalene triangle inscribed in a circle with centre $O$. The altitudes $AD,BE,CF$ meet at the orthocentre $H$. Let $G$ be the reflection of $O$ in $BC$. Draw diameters $EK,FL$ of the circles $(GHE),(GHF)$, respectively.
(a) Suppose $AK,AL$ meet $DE,DF$ at $U,V$, respectively. Prove that $UV\parallel EF$.
(b) The tangents to $(O)$ at $B,C$ meet at $S$. Let $T=DS\cap HG$. Let $M,N$ be the perpendicular projections of $H$ onto the lines $TE,TF$, respectively. Prove that $M,N,E,F$ are concyclic.
Editorial caveat: the source omits a necessary construction condition in (b). Even for an acute scalene triangle, $DS$ and $HG$ can be parallel, so $T$ need not exist. The conclusion below is proved when this intersection is finite; a fully checked counterexample to its automatic existence is given after the proof. The source statement has not been silently amended.
(a) Suppose $AK,AL$ meet $DE,DF$ at $U,V$, respectively. Prove that $UV\parallel EF$.
(b) The tangents to $(O)$ at $B,C$ meet at $S$. Let $T=DS\cap HG$. Let $M,N$ be the perpendicular projections of $H$ onto the lines $TE,TF$, respectively. Prove that $M,N,E,F$ are concyclic.
Editorial caveat: the source omits a necessary construction condition in (b). Even for an acute scalene triangle, $DS$ and $HG$ can be parallel, so $T$ need not exist. The conclusion below is proved when this intersection is finite; a fully checked counterexample to its automatic existence is given after the proof. The source statement has not been silently amended.