Question 93
An arbitrary point $D$ is marked inside the triangle $ABC$. The lines $AD$, $BD$ and $CD$ intersect the sides $BC$, $CA$ and $AB$ at the points $K$, $L$ and $M$ respectively. Let $A_1$, $B_1$ and $C_1$ be the midpoints of the segments $BC$, $CA$ and $AB$, and $X$, $Y$ be the midpoints of the segments $ML$ and $KM$ respectively. The lines $XA_1$ and $YB_1$ intersect at the point $F$.
Prove that $FO \parallel DH$, where $O$ is the circumcenter of the triangle $A_1B_1C_1$ and $H$ is the orthocenter of the triangle $ABC$.
(Mikhail Karpuk)
Prove that $FO \parallel DH$, where $O$ is the circumcenter of the triangle $A_1B_1C_1$ and $H$ is the orthocenter of the triangle $ABC$.
(Mikhail Karpuk)