Prove that the medians of an equilateral triangle are equal.

In the figure, $\mathrm{AC}=\mathrm{DE}, \angle \mathrm{ACB}=\angle \mathrm{EDF}$ and $\mathrm{BD}=\mathrm{CF}$. Prove that $\mathrm{AB}=\mathrm{EF}$. 

In the figure, $\mathrm{AC}=\mathrm{AE}, \mathrm{AB}=\mathrm{AD}$ and $\angle \mathrm{BAD}=\angle \mathrm{EAC}$. Prove that $\mathrm{BC}=\mathrm{DE}$.

In the given figure, $\mathrm{PS}$ is a median and $\mathrm{QL}$ and $\mathrm{RM}$ are perpendiculars drawn from $\mathrm{Q}$ and $\mathrm{R}$ respectively on $\mathrm{PS}$ and $\mathrm{PS}$ produced. Prove that $\mathrm{QL}=\mathrm{RM}.$

In the figure, $\mathrm{QX}$ and $\mathrm{RX}$ are the bisectors of the angles $\mathrm{Q}$ and $\mathrm{R}$ respectively of $△\mathrm{PQR}$. If $\mathrm{XS}\perp \mathrm{QR}$ and $\mathrm{XT}\perp \mathrm{PQ}$, prove that $△\mathrm{XTQ}\cong △\mathrm{XSQ}$.

In the figure, $\mathrm{QX}$ and $\mathrm{RX}$ are the bisectors of the angles $\mathrm{Q}$ and $\mathrm{R}$ respectively of $△\mathrm{PQR}$. If $\mathrm{XS}\perp \mathrm{QR}$ and $\mathrm{XT}\perp \mathrm{PQ}$, prove that $\mathrm{PX}$ bisects the angle $\mathrm{P}$.

Find the values of $\mathrm{x}$ and $\mathrm{y}$ in the figure given below, using congruency of triangles.

Find the values of $\mathrm{x}$ and $\mathrm{y}$ in the figure given below, using congruency of triangles.