In $\Delta ABC$, the bisector of $\angle A$ intersects $BC$ at $D$. A perpendicular from $B$ to $AD$ is drawn intersecting it at $E$. Let a parallel line from $E$ parallel to $AC$ be drawn and intersecting $AB$ and $BC$ at $H$ and $G$ respectively. If $AB = 26$ , $BC = 28$ , $CA = 30$ , find $DG$.
What I Tried: Here is a picture :-
I can see $\Delta EDG \sim \Delta ADC$ . So probably I have to find the lengths of $AD$ and $DC$ and one of $ED$ or $EG$, but how do I do that?
Other than that, I drew perpendiculars from all sides $AB,BC,CA$ and you can find out their lengths from the area of the triangle and base, the area you can get from Heron's Formula. But I got no use of it, I just got the side lengths of the triangle in one way or the other.
Can anyone help me how to find this length of $DG$? I suppose I have to use similarity but I am not getting how to do it.
Extend $BE$ to intersect $AC$ in $F$.
$\triangle ABE \cong \triangle AFE \Rightarrow BE = EF$.
By converse of midpoint theorem in $\triangle ABF$, $H$ is midpoint of $AB$.
Similarly, $G$ is midpoint of $BC$. $BG=14$.
Now $$BD = \dfrac{AB}{AB+AC}\cdot BC = 13$$
$$\therefore \boxed{DG=1}$$