Tricky altitudes

Geometry Level 4

ΔABC \Delta ABC is an isosceles triangle with AC=BC AC = BC and AB=2 AB = 2 . D D is the midpoint of AB AB . The ratio of the radii of it's inscribed circle (radius is r1r_1) and circumscribed circle (radius is r2r_2) is 3:8 3:8 .

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

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

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