$S$ 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,$E$ is a point on the side $CB$ produced of an isosceles triangle $ABC$ with $AB = AC$. If $AD \perp BC$ and $EF \perp AC$,prove that $\Delta ABD \sim \Delta ECF$.

In $\Delta ABC$,$AB = 6\sqrt{3} \text{ cm}$,$AC = 12 \text{ cm}$,and $BC = 6 \text{ cm}$. The angle $B$ is (in $^o$):

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

In the figure,$ABC$ and $DBC$ are two triangles on the same base $BC$. If $AD$ intersects $BC$ at $O$,show that $\frac{\operatorname{ar}(ABC)}{\operatorname{ar}(DBC)} = \frac{AO}{DO}$.

In the figure,$D$ is a point on side $BC$ of $\Delta ABC$ such that $\frac{BD}{CD} = \frac{AB}{AC}$. Prove that $AD$ is the bisector of $\angle BAC$.