Cyclic Quadrilaterals

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

Prove that a cyclic parallelogram is a rectangle.

$\mathrm{A}\mathrm{B}\mathrm{C}$ and $\mathrm{A}\mathrm{D}\mathrm{C}$ are two right triangles with common hypotenuse $\mathrm{A}\mathrm{C}$. Prove that $\mathrm{\angle }\mathrm{C}\mathrm{A}\mathrm{D}=\mathrm{\angle }\mathrm{C}\mathrm{B}\mathrm{D}$.

If circles are drawn taking two sides of a triangle as diameters, prove that the point of intersection of these circles lie on the third side.

Two circles intersect at two points $\mathrm{B}$ and $\mathrm{C}.$ Through $\mathrm{B},$ two line segments $\mathrm{A}\mathrm{B}\mathrm{D}$ and $\mathrm{P}\mathrm{B}\mathrm{Q}$ are drawn to intersect the circles at $\mathrm{A}, \mathrm{D}$ and $\mathrm{P}, \mathrm{Q}$ respectively. Prove that $\mathrm{\angle }\mathrm{A}\mathrm{C}\mathrm{P}=\mathrm{\angle }\mathrm{Q}\mathrm{C}\mathrm{D}.$

If the non-parallel sides of a trapezium are equal, prove that it is cyclic.

If diagonals of a cyclic quadrilateral are diameters of the circle through the vertices of the quadrilateral, prove that it is a rectangle.

$\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{D}$ is a cyclic quadrilateral whose diagonals intersect at a point $\mathrm{E}.$ If $\mathrm{\angle }\mathrm{D}\mathrm{B}\mathrm{C}=70°, \mathrm{\angle }\mathrm{B}\mathrm{A}\mathrm{C}$ is $30°,$ find $\mathrm{\angle }\mathrm{B}\mathrm{C}\mathrm{D}.$ Further, if $\mathrm{A}\mathrm{B}=\mathrm{B}\mathrm{C},$ find $\mathrm{\angle }\mathrm{E}\mathrm{C}\mathrm{D}.$