I think I finally came up with a simple formula in complex numbers for the location of the incenter of a triangle in the complex plane: If a,b,c are the complex numbers which represent the _vectors_ of the triangle sides (a+b+c=0), if R is the (real) circumradius (R=|a||b||c|/(4 Area)), if r is the (real) inradius, if O is the (complex) circumcenter, and if I is the (complex) incenter, then I = O + d (d is a complex number which is the vector from O to I) = O + R*(d/R) = O + R*(a/|a|+b/|b|+c/|c|)/i (i = sqrt(-1)) = O - i*R*(a/|a|+b/|b|+c/|c|) = O - i*R*(abar + bbar + cbar) (abar, bbar, cbar are unit vectors along a,b,c) So we add up the _unit vectors_ along a,b,c, multiply by the circumradius & take a right turn. Note also that d*d' = |d|^2 = R^2 - 2*r*R (Euler's triangle theorem.) Proving Euler's theorem in this fashion required some very deep conversations with Maxima to deal with intermediate expression "swell"; if you're not careful, some of the intermediate expressions can grow to thousands of terms, with no hope of factoring on a laptop.