$A$ point located in the plane of a circle and which is equidistant from every point on the circle is called the ........ of the circle.

The chord which passes through the centre of the circle is called a .......... of the circle.

In the figure,$AOB$ is a diameter of the circle and $C, D, E$ are any three points on the semi-circle. Find the value of $\angle ACD + \angle BED$.

There is one and only one circle passing through all the vertices $A, B$ and $C$ of $\Delta ABC$. This circle is called the ............. of $\Delta ABC$.

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

$A, B$ and $C$ are three points on a circle. Prove that the perpendicular bisectors of $AB, BC$ and $CA$ are concurrent.