Prove that the circle drawn with any side of a rhombus as diameter passes through the point of intersection of its diagonals.

$\mathrm{AC}$and $\mathrm{BD}$are chords of a circle which bisect each other. Prove that $\mathrm{ABCD}$ is a rectangle.

Bisectors of angles $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$of a $∆\mathrm{ABC}$intersect its circumcircle at $\mathrm{D}, \mathrm{E}$ and $\mathrm{F}$, respectively. Prove that the angles of the $∆\mathrm{DEF}$are $90°-\frac{1}{2}\mathrm{A}, 90°-\frac{1}{2}\mathrm{B}$ and $90°-\frac{1}{2}\mathrm{C}$.

Two congruent circles intersect each other at points $\mathrm{A}$and $\mathrm{B}$. Through $\mathrm{A}$ any line segment $\mathrm{PAQ}$is drawn so that $\mathrm{P}$, $\mathrm{Q}$ lie on the two circles. Prove that $\mathrm{BP}=\mathrm{BQ}$.

In any $∆\mathrm{ABC}$, if the angle bisector of $\angle \mathrm{A}$ and perpendicular bisector of $\mathrm{BC}$intersect, prove that they intersect on the circumcircle of the $∆\mathrm{ABC}$.