What have you tried so far? Remember we can give you better hints if you tell us how/where you got stuck.

Here are some hints to get you started:

• 19: Use Law of Sines on $\triangle ACD$, and $\triangle ABD$ to show that $$\dfrac{\sin \angle BAD}{\sin \angle CAD} = \dfrac{3}{4}.$$ Then use it again in $\triangle AGE$ and $\triangle AFG$ to find $\dfrac{EG}{GF}$. At some point you might want to use the identity $\sin(\theta) = \sin(180^\circ - \theta)$.
• 20: Denote the sides of $\triangle ABC$ by $a$, $b$, and $c$, and use the fact that $[ABC] = [ABE] + [BEC]$ (here $[ABC]$ means area of $ABC$), along with the area formula $\dfrac{1}{2}ac\sin(B)$ to show that $BE = \dfrac{2ac}{a+c} \cos\left(\frac{B}{2}\right)$. Use the Angle Bisector Theorem to find expressions for $BD$ and $AE$ in terms of $a$, $b$, and $c$, and use these to find an expression for $\cos\left(\frac{B}{2}\right)$ in terms of $a$, $b$, and $c$. Then use Law of Cosines and Law of Sines.
• 21: Try to find what are $BD$ and $AD$: Let $K$ be a point on $\overline{AB}$ such that $AD = BK$; show that $ECFK$ is a parallelogram and use this to show that $D$ is the midpoint of $\overline{GK}$.