Question 23
Let $(O)$ be a fixed circle and let $B,C$ be fixed points on it. Let $A$ move on $(O)$ so that triangle $ABC$ is acute and scalene. Let $I$ be the midpoint of $BC$, and let $AD,BE,CF$ be the altitudes of triangle $ABC$. Choose $M,N$ on the rays $\overrightarrow{FA},\overrightarrow{EA}$, respectively, so that $FM=CE$ and $EN=BF$. Let $L=MN\cap EF$, and let $G\ne L$ be the second intersection of the circumcircles of triangles $LEN$ and $LFM$.
a) Prove that the circumcircle of triangle $MNG$ always passes through a fixed point.
b) Let $K\ne A$ be the second intersection of $AD$ with $(O)$. On the tangent at $D$ to the circumcircle of triangle $DKI$, choose $P,Q$ so that $GP\parallel AB$ and $GQ\parallel AC$. Let $T$ be the circumcenter of triangle $GPQ$. Prove that $GT$ always passes through a fixed point.
a) Prove that the circumcircle of triangle $MNG$ always passes through a fixed point.
b) Let $K\ne A$ be the second intersection of $AD$ with $(O)$. On the tangent at $D$ to the circumcircle of triangle $DKI$, choose $P,Q$ so that $GP\parallel AB$ and $GQ\parallel AC$. Let $T$ be the circumcenter of triangle $GPQ$. Prove that $GT$ always passes through a fixed point.