In the figure below $\mathrm{ABDC}$ is a cyclic quadrilateral. If $\mathrm{AB}=\mathrm{CD}$, prove that $\mathrm{AD}=\mathrm{BC}$

In the adjoining figure, $ABC$ is an isosceles triangle with $AB=AC.$ If $\angle ABC=50°$, find $\angle BDC, \angle BEC.$

From a point $\mathrm{P}$ outside a circle, with circle $\mathrm{O}$, tangent $\mathrm{PA}$ and $\mathrm{PB}$ are drawn. Prove that $\mathrm{OP}$ is the perpendicular bisector of chord $\mathrm{AB}$. 

The figure given below shows two circles with centres $A,\hspace{0.17em}B$ and a transverse common tangent to these circles meet the straight line $AB$ in $C$. Prove that $AP:BQ=PC:CQ$.

In the figure given below, $PQ$ is a tangent to the circle with centre $O$ and $AB$ is a diameter of the circle. If $QA$ is parallel to $PO$, prove that $PB$ is tangent to the circle.

In the figure given below, two circles with centres $A$ and $B$ touch externally. $PM$ is a tangent to the circle with centre $A$ and $QN$ is a tangent to the circle with centre $B$. If $PM=15 \mathrm{cm},QN=12 \mathrm{cm},PA=17 \mathrm{cm}$ and $QB=13 \mathrm{cm}$, then find the distance between the centres $A$ and $B$ of the circles.