Question 5
A circle $(O)$ and a chord $BC$ are fixed. A point $A$ moves on $(O)$ so that $ABC$ is acute and scalene. Let $I$ be the midpoint of $BC$, and let $AD,BE,CF$ be the altitudes. On rays $FA,EA$, respectively, choose $M,N$ such that $FM=CE$ and $FN = BF$. Suppose $MN$ meets $EF$ at $L$, and circles $(LEN)$ and $(LFM)$ meet again at $G$.
(a) Prove that circle $(MNG)$ always passes through a fixed point.
(b) Suppose $AD$ meets $(O)$ again at $K$. On the tangent at $D$ to circle $(DKI)$, take $P,Q$ such that $GP\parallel AB$ and $GQ\parallel AC$. Let $T$ be the center of circle $(GPQ)$. Prove that line $GT$ always passes through a fixed point.
The supplied scan states FN = BF. The external booklet states EN = BF. The following reference solution applies only to the corrected EN = BF version; it is not a proof of the literal FN = BF statement.
Corrected reference statement (booklet):
A circle $(O)$ and a chord $BC$ are fixed. A point $A$ moves on $(O)$ so that $ABC$ is acute and scalene. Let $I$ be the midpoint of $BC$, and let $AD,BE,CF$ be the altitudes. On rays $FA,EA$, respectively, choose $M,N$ such that $FM=CE$ and $EN = BF$. Suppose $MN$ meets $EF$ at $L$, and circles $(LEN)$ and $(LFM)$ meet again at $G$.
(a) Prove that circle $(MNG)$ always passes through a fixed point.
(b) Suppose $AD$ meets $(O)$ again at $K$. On the tangent at $D$ to circle $(DKI)$, take $P,Q$ such that $GP\parallel AB$ and $GQ\parallel AC$. Let $T$ be the center of circle $(GPQ)$. Prove that line $GT$ always passes through a fixed point.
(a) Prove that circle $(MNG)$ always passes through a fixed point.
(b) Suppose $AD$ meets $(O)$ again at $K$. On the tangent at $D$ to circle $(DKI)$, take $P,Q$ such that $GP\parallel AB$ and $GQ\parallel AC$. Let $T$ be the center of circle $(GPQ)$. Prove that line $GT$ always passes through a fixed point.
The supplied scan states FN = BF. The external booklet states EN = BF. The following reference solution applies only to the corrected EN = BF version; it is not a proof of the literal FN = BF statement.
Corrected reference statement (booklet):
A circle $(O)$ and a chord $BC$ are fixed. A point $A$ moves on $(O)$ so that $ABC$ is acute and scalene. Let $I$ be the midpoint of $BC$, and let $AD,BE,CF$ be the altitudes. On rays $FA,EA$, respectively, choose $M,N$ such that $FM=CE$ and $EN = BF$. Suppose $MN$ meets $EF$ at $L$, and circles $(LEN)$ and $(LFM)$ meet again at $G$.
(a) Prove that circle $(MNG)$ always passes through a fixed point.
(b) Suppose $AD$ meets $(O)$ again at $K$. On the tangent at $D$ to circle $(DKI)$, take $P,Q$ such that $GP\parallel AB$ and $GQ\parallel AC$. Let $T$ be the center of circle $(GPQ)$. Prove that line $GT$ always passes through a fixed point.