The Tiny Triangle

Geometry Level 5

Regular dodecahedron $ABCDEFGHIJKL$ with side length $1$ is drawn. Segments $AF$ , $BG$ , and $CH$ are then drawn.

The area of the tiny little triangle formed from the intersections of these segments can be expressed as $\dfrac{\sqrt{a}-b}{c}$ for positive integers $a,b,c$ . Find the value of $a+b+c$ .