A circle touches all the four sides of a quadrilateral $ABCD$. Prove that $AB+CD=BC+DA$

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, $O$ is the centre of the circle and $x=k°$ then, the measure of $k=$_____.

In the given figure, the measure of $\angle x=$ _____ degree.

The angle between the tangent at a point on a circle and the radius through the point is _____$°$. 

Prove that if a point is marked on each side of a triangle $∆ABC$, then the three Miquel circles (each through a vertex and the two marked points on the adjacent sides) are concurrent at a point $M$ called the Miquel point.

$O$ is the centre of the circle and $x=k°$ then the measure of $k=$ _____

In the given figure, $\angle x=$_____ degree.

In the given figure, find the measurement of the unknown angle in the circle with centre $O$.

If $\angle x=k°$, then $k=$ _____. 

Prove that if $A^{\text{'}},B^{\text{'}},C^{\text{'}}$ be the points on the sides $BC, CA, AB$ of $∆ABC$, respectively, then the circles $A{B}^{\text{'}}{C}^{\text{'}}, A^{\text{'}}B{C}^{\text{'}}$ and $A^{\text{'}}{B}^{\text{'}}C$ pass through a common point. (called the Miquel point)