Simplificar[ p_]  :=  
      Simplify[(p/PolynomialGCD)[p[[1]], p[[2]], p[[3]]]]

Unir[{x1_, y1_, z1_}, {x2_, y2_, z2_}] :=
Simplificar[{Det[(({{y1, z1}, {y2, z2}}))],-Det[(({{x1, z1}, {x2, z2}}))],Det[(({{x1, y1}, {x2, y2}}))]}]

s = ((a + b + c)/2); SA = (((b^2) + (c^2) - (a^2))/2); 
    SB = (((c^2) + (a^2) - (b^2))/2); 
    SC = (((a^2) + (b^2) - (c^2))/2);
CuadradoDistancia[{u_, v_, w_}, {x_, y_, 
            z_}]  := (1/((((u + v + 
                        w))^2) (((x + y + z))^2))) (SA ((((v + w) x - u (y + z)))^2) + SB ((((w + u) y - v (z + x)))^2) + 
              SC ((((u + v) z - w (x + y)))^2));

Area[{x1_, y1_, z1_}, {x2_, y2_, z2_}, {x3_, y3_, z3_}] := 
(1/((x1 + y1 + z1) (x2 + y2 + z2) (x3 + y3 + 
                    z3)))Det[(({{x1, y1, z1}, {x2, y2, z2}, {x3, y3, z3}}))];

ptI = {a, b, c};
ptD = {s - c, -b, 0};
ptE = {s - b, 0, -c};
ptF = {-a, s - c, 0};
ptG = {0, s - a, -c};
ptN = {-a, 0, s - b};
ptM = {0, -b, s - a};

rtDE = Unir[ptD, ptE];
rtFG = Unir[ptF, ptG];
rtMN = Unir[ptM, ptN];

ptX = Unir[rtMN,  rtFG];
ptY = Unir[rtMN,  rtDE];
ptZ = Unir[rtFG, rtDE];

Factor[CuadradoDistancia[ptI, ptX]]
Factor[CuadradoDistancia[ptI, ptY]]
Factor[CuadradoDistancia[ptI, ptZ]]

Factor[Area[ptX, ptY, ptZ]]

ptA = {1, 0, 0};
ptB = {0, 1, 0};
ptC = {0, 0, 1};
Simplify[
      Det[{Unir[ptA, ptX], Unir[ptB, ptY], Unir[ptC, ptZ]}]]

coA = {((4 (Sqrt[35]))/3), (13/3)}; coB = {0, -6}; 
    coC = {0, 0};
{u, v, 
          w}  = (Unir[Unir[ptA, ptX], Unir[ptC, ptZ]]  /. {a -> 6,  
              b -> 9, c -> 13});
 N[((u coA +  v coB +  w coC)/(u + v + w)), 
        12][[1]]

Este punto no se halla en la enciclopedia de Kimberling