In $\Delta ABC$,$m \angle B = 90^{\circ}$ and $\overline{AB} \cong \overline{BC}$. Find the ratio $AB : AC$.

In $\Delta ABC$,$A-M-B$,$A-N-C$ and $\overline{MN} \parallel \overline{BC}$. If $AB = 10.8$,$MB = 4.5$ and $AC = 4.8$,find $AN$.

$O$ is any point inside a rectangle $ABCD$ (see Figure). Prove that $OB^{2} + OD^{2} = OA^{2} + OC^{2}$.

In $\Delta ABC$,$A-P-B$,$A-Q-C$ and $\overline{PQ} \parallel \overline{BC}$. If $AP = 8x-7$,$PB = 5x-3$,$AQ = 4x-3$ and $QC = 3x-1$,find the value of $x$.

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 $\overline{PQ} \parallel \overline{AB}$.

The vertices of a $\Delta ABC$ are $A(4, 6)$,$B(1, 5)$,and $C(7, 2)$. $A$ line is drawn to intersect sides $AB$ and $AC$ at $D$ and $E$ respectively,such that $\frac{AD}{AB} = \frac{AE}{AC} = \frac{1}{4}$. Calculate the area of $\Delta ADE$ and compare it with the area of $\Delta ABC$.