Does the median bisect the angles?

Circles $C_1$ and $C_2,$ of radii $r$ and $R$ respectively, touch each other as shown in the figure. The line $l$ which is parallel to the line joining the centres of $C_1$ and $C_2,$ is tangent to $C_1$ at $P$ and intersects $C_2$ at $A,B$. If $R^2=2r^2,$ then $\angle AOB$ equals-

In the figure given below, $CDE$ is an equilateral triangle formed on a side $CD$ of a square $ABCD$. Show that, $△ADE\cong △BCE$and hence, $AEB$ is an isosceles triangle.

In the adjoining, $AD, BE$ and $CF$ are altitudes of $△ABC$. If $AD=BE=CF$, prove that $ABC$ is an equilateral triangle.

Restate the following statement with appropriate condition, so that it become true statement.

In every triangle, a median is also an angle bisector.