Question 44
Let $ABC$ be a right triangle with right angle at $A$, and let $AD$ be its altitude from $A$ to $BC$. On the ray $[AD$, take points $E$ and $H$, such that $AE = AC$ and $AH = AB$. Construct squares $AEFG$ and $AHJI$, such that $C$ lies inside $AEFG$, and $B$ lies inside $AHJI$. Let $K = AC \cap EG$, $L = AB \cap IH$, $N = IL \cap GK$, $M = IB \cap GC$. Prove that:
a) $LK \parallel BC$
b) the points $A, N, M$ are collinear.
a) $LK \parallel BC$
b) the points $A, N, M$ are collinear.