# Tricky altitudes

Geometry Level 4

$$\Delta ABC$$ is an isosceles triangle with $$AC = BC$$ and $$AB = 2$$. $$D$$ is the midpoint of $$AB$$. The ratio of the radii of it's inscribed circle (radius is $$r_1$$) and circumscribed circle (radius is $$r_2$$) is $$3:8$$.

Two different triangles can be formed under these conditions. In one triangle the square of length $$CD$$ is an integer $$a$$, in the other the square of length $$CD$$ can be expressed as $$\frac{m}{n}$$, where $$m$$ and $$n$$ are coprime positive integers.

What is the value of $$a + m + n$$?

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