If three equal circles touch each other, then prove that their centres are the vertices of an equilateral triangle.

$PQ$ is a diameter of a circle and $A$ is a point on the circumference. $PR$ is perpendicular to the tangent at $A$. Prove that $PA$ bisects $\angle QPR$.

$AB$ is the diameter of a circle. The tangent at $C$ (on the circle) meets $AB$ produced at $D$. Prove that $\angle BDC+2\angle BCD=1$ right angle.

$AB$ is a diameter of a circle with centre $O$. $PAQ$ and $XBY$ are two tangents of the circle at $A$ and $B$. Prove that $PQ\parallel XY$.

$AB$ is the diameter of a circle with centre $O$. The chord $CD$ of the circle is parallel to the tangent at $A$. Prove that the diameter $AB$ is the perpendicular bisector of the chord $CD$.

$ABCD$ is a quadrilateral circumscribed about a circle with centre $O$. Prove that,

$\left(i\right) AB+CD=BC+DA$

$\left(ii\right) \angle AOB+\angle COD=2$ right angles. 

The parallelogram circumscribed about a circle must be a rhombus. Prove it.

The rectangle circumscribed about a circle is a square. Prove it.

$ABC$ is an isosceles triangle of which $AB=AC$. A circle inscribed in the triangle touches the sides $AB, BC$ and $CA$ at $D, E$ and $F$ respectively. Prove that $BE=CE$.