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 \(a\).