# Distance for 3D Objects

Geometry Level 4

A sphere is inscribed in a regular tetrahedron with side length of $$2$$ such that it is tangent to each of the tetrahedron's four faces. Let points $$P$$ and $$Q$$ be two of the points of tangency. If $$d$$ is the distance between $$P$$ and $$Q$$, then $$d^6$$ can be expressed as $$\dfrac{m}{n}$$, where $$m$$ and $$n$$ are positive, coprime integers. Find $$m+n$$.

×