I need a short way to solve following problem:
Suppose that length of chord $AB=5\operatorname{cm}$,$AC=7\operatorname{cm}$ and $BC=8\operatorname{cm}$, we know that $D$ is midpoint of arc $BC$ and chord $AD$ divides $BC$ into two equal parts (let intersection point be $K$) so $BK=KC=4$, we are going to find $AK$ and $KD$.
First of all I know characters of chords intersection , which means that $AK\cdot KD=BK \cdot KC$, sure we can find by cosine theorem $AK$(we know all length,we can find any angle and then repeat usage of cosine)but because $D$ is midpoint of arc $BC$ and also $AD$ is median, I doubt that AD is the bisector, diameter or something like that.
Thanks a lot.