If the altitude from one vertex of a triangle bisects the opposite side. Prove that the triangle is isosceles.

If $∆APC$ is congruent to $∆PQR$, then find the length of $QR$.

In the given figure, $P$ is a point equidistant from two lines $l$ and $m$ intersecting at point $A$. Prove that $∆ABP$ and $∆ACP$ are congruent and length of $AC$ is $8 \mathrm{cm}$.

In the given figures, $AB=DE, BC=EF$ and median $AP=$median $DQ.$ Prove that $\angle B=\angle E.$

It is given that $DE=EF, BE=EC, ED\perp AB$ and $EF\perp AC$. Prove that $AB=AC$.