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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