Show that a point is a midpoint of a side of a triangle

Trivial by picture:

$BEA$ is a right triangle and by Thales' theorem $\widehat{BGH}=\widehat{BCA}$, $\widehat{BHG}=\widehat{BAC}$.
Angle chasing gives $\widehat{HEA}=\widehat{EAC}=\widehat{HAE}$, hence $HA=HB=HE$.

HINT :

We have $\angle{HAE}=\angle{HEA}$, so $\triangle{HAE}$ is an isosceles triangle with $HA=HE$.

Now prove that $\triangle{HBE}$ is an isosceles triangle with $HB=HE$.

Let $\alpha = \frac12 \angle CAB$. We have $\alpha = \angle AEH$ (alternate interior angles), so $\triangle AEH$ is isosceles of vertex $H$, so $HA = HE$.

Let $\beta = \angle HBE$, $\gamma = \angle EBD$ and $\theta = \angle EDG$. We have:

$$\beta + \gamma = \pi- (\angle ACB + 2 \alpha) = \pi - \angle EGD - 2\alpha = \pi - (\pi - (\alpha + \theta)) - 2\alpha = \theta - \alpha = (\pi - \angle EDB) - \alpha = \pi - (\pi/2 - \gamma) - \alpha = \pi/2 + \gamma - \alpha$$

Hence, $\beta = \pi/2 - \alpha = \angle HEB$, so $\triangle HEB$ is isosceles of vertex $H$ and $HE = HB$.

Therefore, $HB = HA$