Show that sphere passes through 9point circles of faces of a tetrahedron

Question

O is the origin and A, B, C are points (4a,4b,4c),(4b,4c,4a),(4c,4a,4b) show that sphere x^2+y^2+z^2-2(x+y+z)(a+b+c)+8(bc+ca+ab)=0 passes through nine point circles of faces of tetrahedron OABC

Answer 1

It suffices to find the sphere (if any) that passes through the midpoints of every edge (6 in total). (Do you see why?)

We can easily (manually) verify that these 6 points,
3 of which are cyclic permutations of $(2a, 2b, 2c)$,
3 of which are cyclic permutations of $(2a+2b, 2b+2c, 2c+2a)$,
satisfy the equation

$$ x^2 + y^2 + z^2 - 2 (x+y+z) ( a+b+c) + 8 (bc+ca+ab) = 0. $$