A Ball Above the Planes

Geometry Level 5

$\begin{eqnarray} && x + 3 y + 2 z = 5 \\ && 2x - y + z = 5 \\ && x + y - z = 5\end{eqnarray}$

In the following, it is assumed the that the $$xy$$ plane lies horizontally, and that the $$z$$ axis points upward.

Three planes are given as shown above.

A ball of radius 10 units is placed above the three planes such that it is tangent to all three of them.

What is the $$z$$ coordinate of the top of the ball?

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