(* For footnote 23 in the 12th edition of my book, regarding computing gmr vectors. The two equations that must be simultaneously satisfied by the XYZ coordinate (x3,y3,z3): *)
(* written June 2017 by Kurt Johmann *)
Solve[{((x3 - x1)/(x2 - x1)) == ((y3 - y1)/(y2 - y1)) == ((z3 - z1)/(z2 - z1)), L == Sqrt[(x3 - x1)^2 + (y3 - y1)^2 + (z3 - z1)^2]}, {x3,y3,z3}, Reals]