In $\Delta XYZ$,the bisector of $\angle Y$ intersects $\overline{XZ}$ at $M$. Then,which of the following holds true?

If a line intersects sides $AB$ and $AC$ of a $\Delta ABC$ at $D$ and $E$ respectively and is parallel to $BC$,prove that $\frac{AD}{AB} = \frac{AE}{AC}$ (see Figure).

Point $O$ lies in the interior of $\Delta ABC$. The bisectors of $\angle AOB, \angle BOC$ and $\angle COA$ intersect $\overline{AB}, \overline{BC}$ and $\overline{CA}$ at $D, E$ and $F$ respectively. Prove that $AD \times BE \times CF = DB \times EC \times FA$.

In $\Delta PQR$,$m \angle Q = 90^{\circ}$ and $\overline{QS}$ is an altitude. If $PS - SR = 10$ and $PQ^2 - QR^2 = 260$,then $PR = \dots$

$A$ girl of height $90\, cm$ is walking away from the base of a lamp-post at a speed of $1.2\, m/s$. If the lamp is $3.6\, m$ above the ground,find the length of her shadow after $4\, seconds$.

The diagonals of $\square XYZW$ intersect at right angles. Prove that $XY^{2} + ZW^{2} = YZ^{2} + XW^{2}$.