In $\Delta ABC$, $M$ and $N$ are the midpoints of $\overline{AB}$ and $\overline{AC}$ respectively. If the area of $\Delta ABC$ is $90$, find the area of $\Delta AMN$.

In $\Delta ABC$,if $\ldots \ldots \ldots \ldots$,then by Apollonius' theorem,$AB^{2} + AC^{2} = 2(AD^{2} + BD^{2})$ holds good.

$A$ triangle whose sides measure ............ is a right-angled triangle.

In $\Delta ABC$,$m \angle B = 90^{\circ}$ and $\overline{BM}$ is an altitude to the hypotenuse $AC$. If $AM = 7$ and $CM = 9$,then $BC = \ldots$.

In $\Delta XYZ$,$\overline{XM}$ is a median. If $XY = 20$,$XZ = 21$ and $XM = 14.5$,find $YZ$.

In $\Delta ABC$,$\overline{AD}$ is a median. The bisectors of $\angle ADB$ and $\angle ADC$ intersect $\overline{AB}$ and $\overline{AC}$ at $E$ and $F$ respectively. Prove that $\overline{EF} \parallel \overline{BC}$.