Determining an angle bisector in a circle

Question

Let $K$ be a circle with center $B$ and $A,C,D$ points on the circle. Further let $\overline{BE}$ be the perpendicular bisector of $\overline{AC}$.

Then it happens to be that the line $\overline{DE}$ is the angle bisector of $\sphericalangle CDA$, but how does one proof this?

Answer 1

Angles subtended by an arc at the center of a circle are double those at the circumference:

$$\angle ADE = {1\over 2}\angle ABE = {1\over 2}\angle EBC = \angle EDC$$