Define Simson line?

On the circle with center $O$, points $A,B$ are such that $OA = AB$. A point $C$ is located on the tangent at $B$ to the circle such that $A$ and $C$ are on the opposite sides of the line $OB$ and $AB=BC$. The line segment $AC$ intersects the circle again at $F$. Then the ratio $\angle BOF:\angle BOC$ is equal to -

Suppose $S_1 \mathrm{a}\mathrm{n}\mathrm{d} S_2$ are two unequal circles; $AB$ and $CD$ are the direct common tangents to these circles. A transverse common tangent $PQ$ cuts $AB$ in $R$ and $CD$ in $S$. If $AB=10$ units, then $RS$ is -

In the given figure, ray $OC$ is the bisector of $\angle AOB$ and $OD$ is the ray opposite to $OC$. Show that $\angle AOD=\angle BOD$.