How many subsidiary triangles does a Sriyantra have?

$O$ is a point in the interior of a square $ABCD$ such that $\triangle OAB$ is an equilateral triangle. Show that $\triangle OCD$ is an isosceles triangle.

$ABC$ is a right-angled triangle with $AB = AC$. The bisector of $\angle A$ meets $BC$ at $D$. Prove that $BC = 2 AD$.

$AD$ is a median of the triangle $ABC$. Is it true that $AB + BC + CA > 2AD$? Give reason for your answer.

The line segment joining the mid-points $M$ and $N$ of the parallel sides $AB$ and $DC$ respectively of a trapezium $ABCD$ is perpendicular to both the sides $AB$ and $DC$. Prove that $AD = BC$.

In a right triangle,prove that the line segment joining the mid-point of the hypotenuse to the opposite vertex is half the length of the hypotenuse.