In $\Delta ABC$,$M$ lies on $AB$ and $N$ lies on $AC$ such that $\overline{MN} \parallel \overline{BC}$. If $\frac{AM}{MB} = \frac{3}{4}$ and $AC = 21$,find $AN$.

In an equilateral triangle $ABC$,$D$ is a point on side $BC$ such that $BD = \frac{1}{3} BC$. Prove that $9 AD^{2} = 7 AB^{2}$.

In $\Delta PQR$,$m \angle P = 90^{\circ}$ and $\overline{PM}$ is an altitude. If $QM = 2x^{2}$ and $RM = 8x^{2}$,then $PM = \ldots \ldots$

The diagonals of convex quadrilateral $PQRS$ intersect at right angles. Prove that $PQ^{2} + RS^{2} = PS^{2} + QR^{2}$.

In $\Delta XYZ$,the midpoints of the sides are $A, B$ and $C$. In $\Delta ABC$,the midpoints of the sides are $P, Q$ and $R$. If the area of $\Delta ABC$ is $24$,find the area of $\Delta XYZ$ and $\Delta PQR$.

In $\Delta ABC$,points $D$ and $E$ lie on $\overline{AB}$ such that $A-D-E-B$ and $AD = EB$. $P \in \overline{AC}$ and $Q \in \overline{BC}$ such that $\overline{DP} \parallel \overline{BC}$ and $\overline{EQ} \parallel \overline{AC}$. Prove that $\overline{PQ} \parallel \overline{AB}$.