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

Sides $AB$ and $AC$ and median $AD$ of a triangle $ABC$ are respectively proportional to sides $PQ$ and $PR$ and median $PM$ of another triangle $PQR$. Show that $\Delta ABC \sim \Delta PQR$.

In the figure,$ABC$ is a triangle in which $\angle ABC < 90^{\circ}$ and $AD \perp BC$. Prove that $AC^{2} = AB^{2} + BC^{2} - 2BC \cdot BD$.

In the figure,two chords $AB$ and $CD$ of a circle intersect each other at point $P$ (when produced) outside the circle. Prove that $\Delta PAC \sim \Delta PDB$.

In the figure,$ABD$ is a triangle right-angled at $A$ and $AC \perp BD$. Show that $AB^2 = BC \cdot BD$.

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