Question 87
Let $H$ be the orthocenter of an acute triangle $ABC$, points $A_1$, $B_1$, $C_1$ be the feet of the altitudes $AA_1$, $BB_1$, $CC_1$, respectively. Let points $D$, $K$, $L$, $M$, $N$ be the midpoints of the segments $AC$, $BC_1$, $BA_1$, $HA_1$, $HC_1$, respectively, and $P$, $Q$, $R$ be the intersection points of the lines $AL$ and $CK$, $AN$ and $CK$, $AL$ and $CM$, respectively.
Prove that the line $DQ$ touches the circumcircle of the triangle $PQR$.
Prove that the line $DQ$ touches the circumcircle of the triangle $PQR$.