2020 HMMT November General Round Problem 6

A sphere is centered at a point with integer coordinates and passes through the three points (2,0,0), (0,4,0), (0,0,6), but not the origin (0,0,0). If r is the smallest possible radius of the sphere, compute r^2.

Answers are checked by value, so any equivalent form is accepted: 1/2, \frac{1}{2} and 0.5 all count as the same answer.

Full credit goes to HMMT for authoring these problems. This problem is from the November 2020 contest; the official solution is available on the HMMT archive. HMMT is not affiliated with or endorsing TopsOJ in any way.