$S$ and $T$ are points on sides $PR$ and $QR$ of $\Delta PQR$ such that $\angle P = \angle RTS$. Show that $\Delta RPQ \sim \Delta RTS$.

In the figure,$AD$ is a median of a triangle $ABC$ and $AM \perp BC$. Prove that:
$AB^{2} = AD^{2} - BC \cdot DM + \left(\frac{BC}{2}\right)^{2}$

$PQR$ is a triangle right-angled at $P$ and $M$ is a point on $QR$ such that $PM \perp QR$. Show that $PM^{2} = QM \cdot MR$.

Two poles of heights $6 \, m$ and $11 \, m$ stand on a plane ground. If the distance between the feet of the poles is $12 \, m$,find the distance between their tops.

$E$ is a point on the side $AD$ produced of a parallelogram $ABCD$ and $BE$ intersects $CD$ at $F.$ Show that $\Delta ABE \sim \Delta CFB$.

State which pairs of triangles in the figure are similar. Write the similarity criterion used by you for answering the question and also write the pairs of similar triangles in the symbolic form.