Kinda Familiar... Ah! A Theorem

In the diagram, a circle $(I; r)$ is internally tangent to a circle $(O; R)$ .

Point $M$ lies on $(O)$ . Draw tangents $MD, ME$ to circle $(I)$ , which intersects $(O)$ at $B,C$ , respectively.

$DE$ intersects $IM$ at $K$ .

If we have $R = 7, r = 3, \angle BMC = 45^\circ$ , then $2OK^2$ can be written as:

$2OK^2 = a-b\sqrt{c}$

where $a, b$ and $c$ are positive integers with $c$ square-free.

Find $a+b+c$ .

Details and assumptions:

This problem is based on a geometric theorem, but it hasn't been wiki-ed yet.