Right Triangles In Regularity (Advanced 1)

Geometry Level 4

As shown in the figure, \(K\) is the midpoint of one side \(AC\) of the equilateral triangle \(\Delta{ABC}\), which has a RIGHT angle \(\angle{MKN}\) inside with two intersections \(M\) and \(N\) with \(AB\) and \(BC\), respectively.

Given \(AM=6\) and \(CN=\frac{4}{5}\), so that \(MN\) can be found in length as \(\dfrac{b}{a}\sqrt{c}\) with \(a\) and \(b\) coprime positive integers and \(c\) square-free. Also we can find \(AC\) in length as a positive integer \(d\).

Find \(a+b+c+d\).

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