State whether each of the following statements is true or false:
$(1)$ $A$ piece of a circle between two points on a circle is called an arc of the circle.
$(2)$ The length of the complete circle is called its circumference.

$A$ circular park of radius $20\,m$ is situated in a colony. Three boys Ankur,Syed and David are sitting at equal distance on its boundary,each having a toy telephone in his hands to talk with each other. Find the length of the string of each phone.

In the figure,$AB$ and $CD$ are two chords of a circle intersecting each other at point $E$. Prove that $\angle AEC = \frac{1}{2}$ (angle subtended by arc $CXA$ at the centre $+$ angle subtended by arc $DYB$ at the centre).

In the figure,$O$ is the centre of the circle,and $\angle BCO = 30^{\circ}$. Find $x$ and $y$.

If arcs $AXB$ and $CYD$ of a circle are congruent,find the ratio of $AB$ and $CD.$

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