Prove that the triangle formed by joining the midpoints of an isosceles triangle is also an isosceles triangle.

In the figure, $AD=BC$, $AC=BD$. Prove that $△PAB$ is an isosceles triangle.

In the figure, $AB=AC, BD=CD$. Prove that $\angle ABD=\angle ACD$.

In the figure, $AB=AC, BD=CD$. Prove that $\angle ABD=\angle ACD$.

In the figure, $△ABC$ is an isosceles triangle in which $AB=AC$. The side $BA$ is produced to $D$ such that $AB=AD$. Prove that $\angle BCD$ is a right angle.

If the bisector of the vertical angle of a triangle bisects the base also, the triangle is isosceles.