One theorem to rule them all... do you know what it is?

Geometry Level 3

ABC is a triangle where AC is of length 13, and AB is of length 15. Point D is constructed on side CB such that the ratio $CD:DB=\frac{1}{2}$ . Let the cevian AD be of length 6. If CD can be expressed as $\frac{\sqrt{a}}{\sqrt{6}}$ , find $a$ .