The longest chord of a circle is 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 $\angle ACD + \angle BED$.

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).

Write True or False and justify your answer in each of the following:
Two congruent circles with centres $O$ and $O^{\prime}$ intersect at two points $A$ and $B$. Then $\angle AOB = \angle AO^{\prime}B$.

If two equal chords of a circle intersect,prove that the parts of one chord are separately equal to the parts of the other chord.

$AB$ and $AC$ are two chords of a circle with centre $P$. If the bisector of $\angle BAC$ passes through the centre $P$,prove that $AB = AC$.