Question 54
Let $P$ and $Q$ be the midpoints of the sides $BC$ and $CD$, respectively, of a rectangle $ABCD$. Let $K$ and $M$ be the points of intersection of the line $PD$ with $QB$ and $QA$, respectively, and let $N$ be the point of intersection of the lines $PA$ and $QB$.
Let $X, Y, Z$ be the midpoints of the line segments $AN, KN, AM$, respectively. Let $\ell_{1}$ be the line passing through $X$ and perpendicular to $MK$, $\ell_{2}$ be the line passing through $Y$ and perpendicular to $AM$, $\ell_{3}$ be the line passing through $Z$ and perpendicular to $KN$. Show that $\ell_{1}, \ell_{2}, \ell_{3}$ are concurrent.
Let $X, Y, Z$ be the midpoints of the line segments $AN, KN, AM$, respectively. Let $\ell_{1}$ be the line passing through $X$ and perpendicular to $MK$, $\ell_{2}$ be the line passing through $Y$ and perpendicular to $AM$, $\ell_{3}$ be the line passing through $Z$ and perpendicular to $KN$. Show that $\ell_{1}, \ell_{2}, \ell_{3}$ are concurrent.