Inspheres in Irregular Tetrahedron

Geometry Level 5

An irregular tetrahedron has four vertices:

$$A (0, 0, 0)$$ , $$B (100, 0, 0)$$, $$C (40, 60, 0)$$ and $$D (60, 40, 100)$$

There exists a sphere inside the tetrahedron that is tangent to all of its four faces. In addition there exists four secondary spheres that are tangent to the first sphere and the three planes that intersect at each of the four vertices.

Find the sum of the radii of all five spheres.

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