TITLE: Show that a point is a midpoint of a side of a triangle
QUESTION [6 upvotes]: In $\Delta ABC$, the bisector of $\angle A$ intersects $BC$ at $D$. The perpendicular to $AD$ from $B$ intersects $AD$ at $E$. The line through $E$ parallel to $AC$ intersects $BC$ at $G$, and $AB$ at $H$.
Prove that $H$ is the mid-point of $AB$.
I have no idea on how to prove this question. Any hint would be much appreciated.

REPLY [2 votes]: We need to consider only the triangle $ABE$ with the point $H$ on $AB$ such that $\angle HEB=\angle EAC=\angle BAE=\theta$. We have $\angle BHE=2\theta$ and $\angle BEH=\frac12\pi-\theta$; so $\angle HBE=\frac12\pi-\theta$. Hence triangle $BHE$ is isosceles, as is $EHA$, with $|HB|=|HE|=|HA|.$