Edit: Nope, nevermind, two of the points both see the medium and small circles!
q3 = (r1 * p2 - r2 * p1) / (r1 - r2)
( r2 p3 - r3 p2 ) ( r3 p1 - r1 p3 ) ( r1 p2 - r2 p1 )
( r2 - r3 ), ( r3 - r1 ), ( r1 - r2 )
A somewhat neater but more advanced proof, follows by making the positions pi also homogeneous and noting that (using Einstein summation notation):
0 = ε_ijk rj rk
qi = ε_ijk rj pk
ri qi = ε_ijk ri rj pk = ε_jik rj ri pk = -ε_ijk ri rj pk = 0.
On the other end it was a Projective Geometry course and at the end we used a lot the General Linear Group, in particular 4x4 matrices for bilinear forms, so it was like warming up with Geometry to arrive to the Algebra tools.