Simplificar[{u_, v_, w_}]  :=  
      Simplify[{u, v, w}/PolynomialGCD[u, v, w]]
Unir[{x1_, y1_, z1_}, {x2_, y2_, z2_}] :=
Simplificar[{Det[(({{y1, z1}, {y2, z2}}))],-Det[(({{x1, z1}, {x2, z2}}))],Det[(({{x1, y1}, {x2, y2}}))]}]
Punto[r_, s_]  :=  
        Unir[r, s];
Recta[P_, Q_]  :=  
        Unir[P, Q];

ptA = {1, 0, 0}; ptB = {0, 1, 0}; ptC = {0, 0, 1};

ptP1 = {u, v, w}; 
ptA1 = {0, v, w};
ptB1 = {u, 0, w};
ptC1 = {u, v, 0};

ptA2 = {0, w, v};
ptB2 = {w, 0, u};
ptC2 = {v, u, 0};

Det[{
Recta[ptA, ptA2], 
Recta[ptB, 
          ptB2], 
Recta[ptC, ptC2]}]

ptP2  =  Punto[Recta[ptA, ptA2], Recta[ptB, ptB2]]

ptA3 = 
      Punto[Recta[ptB1, ptC2], Recta[ptB2, ptC1]]
ptB3 = 
      Punto[Recta[ptC1, ptA2], Recta[ptC2, ptA1]]
ptC3 = Punto[Recta[ptA1, ptB2], Recta[ptA2, ptB1]]

Det[{ptA3, ptB3, ptC3}]

A3B3C3 = Recta[ptA3, ptB3]

Simplify[Tr[A3B3C3 * ptP1]]
Simplify[Tr[A3B3C3 * ptP2]]

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));

CuadradoRazonDoble[P_, Q_, R_, S_]  :=  
Divide[
        CuadradoDistancia[P, R]/CuadradoDistancia[Q, R], 
        CuadradoDistancia[P, S]/CuadradoDistancia[Q, S]]

Factor[
      CuadradoRazonDoble[ptA3, ptB3, ptC3, ptP1]/CuadradoRazonDoble[ptA3, ptB3, ptC3, ptP2]]

Factor[Tr[ptA3]]