Question: [8 points] Consider triangle ????ABC with ?bar (AB)~?b=ar (AC).a) Let M and N be the midpoints of ?bar (AC) and ?bar (AB), respectively. Prove that ?bar (BM)~?b=ar (CN).b) Let D and E be the intersections of the angle bisectors of ??ABC and ??ACB with ?bar (AC) and ?bar (AB),respectively. Prove that ?bar (BD)~?b=ar (CE).c) Let F and G be the

$[8$ points$]$ Consider triangle $\frac{?}{{?}_{\frac{?}{?}}\text{A}\text{B}\text{C}}$ with $\frac{?}{\text{b}\text{a}\text{r}}(AB)$~$\frac{?}{b}=ar(AC).$

a$)$ Let $M$ and $N$ be the midpoints of $\frac{?}{\text{b}\text{a}\text{r}}(AC)$ and $\frac{?}{\text{b}\text{a}\text{r}}(AB),$ respectively. Prove that $\frac{?}{\text{b}\text{a}\text{r}}(BM)$~$\frac{?}{b}=ar(CN).$

b$)$ Let $D$ and $E$ be the intersections of the angle bisectors of $\frac{?}{{?}_{A}BC}$ and $\frac{?}{{?}_{A}CB}$ with $\frac{?}{\text{b}\text{a}\text{r}}(AC)$ and $\frac{?}{\text{b}\text{a}\text{r}}(AB),$

respectively. Prove that $\frac{?}{\text{b}\text{a}\text{r}}(BD)$~$\frac{?}{b}=ar(CE).$

c$)$ Let $F$ and $G$ be the feet of the perpendiculars from $B$ and $C$ to $\frac{?}{\text{b}\text{a}\text{r}}(AC)$ and $\frac{?}{\text{b}\text{a}\text{r}}(AB),$ respectively. Can you

prove that $\frac{?}{\text{b}\text{a}\text{r}}(BF)$~$\frac{?}{b}=ar(CG)?$ If so$,$ do it$.$ If not, explain why not.