# Kinda Familiar... Ah! A Theorem

Geometry Level 5

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.

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