In $\Delta ABC$,the bisector of $\angle A$ intersects $\overline{BC}$ at $D$. If $AB : AC = 3 : 4$ and $BC = 14$,find $BD$.

In $\square ABCD$,$P \in \overline{AB}$,$Q \in \overline{BC}$,$R \in \overline{CD}$,and $S \in \overline{AD}$ such that $\frac{AP}{AB} = \frac{CQ}{BC} = \frac{CR}{CD} = \frac{AS}{AD} = \frac{1}{3}$. Prove that $\square PQRS$ is a parallelogram.

$BL$ and $CM$ are medians of a triangle $ABC$ right-angled at $A$. Prove that $4(BL^2 + CM^2) = 5BC^2$.

In $\Delta ABC$,$m \angle B = 90^{\circ}$ and $\overline{BM}$ is a median. If $AB = 15$ and $BC = 20$,find $BM$.

In a rhombus $ABCD$,prove that $AC^{2} + BD^{2} = 4AB^{2}$.

Below are given the measures of sides $\overline{PQ}$,$\overline{QR}$ and $\overline{PR}$ of $\Delta PQR$. In each case,determine whether $\Delta PQR$ is a right-angled triangle or not. If it is a right-angled triangle,state which angle is a right angle: $PQ = 8, QR = 6, PR = 12$.