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

The "One Theorem to Rule Them All" is Stewart's Theorem. It is a relatively complex formula that is really nifty to hold on to. By letting CD be $x$ , and DB be $2x$ , plugging that into the formula produces $x^{2}=\frac{455}{6}$ . Thus $a=455$ . It's a really nifty theorem that I have seen on tons of AIME problems. :D

Let the ratio be x:2x Then we all have to do is put cosine law for theta and pie-theta.. Add the equations to become zero and then find the value of x.. It comes x^2 = 455/6

You can use The Stewart's Theorem

draw altitude AH;let CH=m , CD=x; use Pythagorean theorem: $AD^{2}$ = $AH^{2}$ + $HD^{2}$ ; $AC^{2}$ = $AH^{2}$ + $CH^{2}$ ; =>(AC-AD)(AC+AD)=(CH-HD)(CH+HD) ; =>133=(2m-x)x=>133=2mx- $x^{2}$ =>399=6mx-3 $x^{2}$ (1 ) ; do the same thing with AC and AB =>56=(3x-2m)3x=>56=9 $x^{2}$ -6mx (2 ) ; plus (1 ) with (2 ) =>455=6 $x^{2}$ => $x^{2}$ =455/6 .