 reserve A, B, C, A1, B1, C1, A2, B2, C2, C3 for Point of TOP-REAL 2,
  lambda, mu, nu, alpha, beta, gamma for Real,
  X, Y, Z for Subset of TOP-REAL 2;

theorem
  Th21: (A, B, C is_a_triangle & A1 = (1 - lambda) * B + lambda * C & B1 = (1
  - mu) * C + mu * A & C1 = (1 - nu) * A + nu * B & lambda <> 1 & mu <> 1 &
  nu <> 1 & ((1 - mu) + lambda * mu) <> 0 & ((1 - lambda) + nu * lambda) <> 0
  & ((1 - nu) + mu * nu) <> 0) implies ((lambda / (1 - lambda)) * (mu / (1 -
  mu)) * (nu / (1 - nu)) = 1 iff (ex A2 st A, A1, A2 are_collinear & B, B1, A2
  are_collinear & C, C1, A2 are_collinear))
proof
  set q1 = ((1 - mu) + lambda * mu);
  set q2 = ((1 - lambda) + nu * lambda);
  set q3 = ((1 - nu) + mu * nu);
  assume that A1: A, B, C is_a_triangle and A2: A1 = (1 - lambda) * B +
  lambda * C and A3: B1 = (1 - mu) * C + mu * A and A4: C1 = (1 - nu) * A +
  nu * B and A5: lambda <> 1 and A6: mu <> 1 and A7: nu <> 1 and A8: q1 <> 0
  and A9: q2 <> 0 and A10: q3 <> 0;
  A11: C, A, B is_a_triangle by A1;
  A12: B, C, A is_a_triangle by A1;
  consider C2 such that A13: A, A1, C2 are_collinear and A14: B, B1, C2
  are_collinear by Lm3,A1,A2,A3,A8;
  consider B2 such that A15: C, C1, B2 are_collinear and A16: A, A1, B2
  are_collinear by Lm3,A11,A4,A2,A9;
  consider A2 such that A17: B, B1, A2 are_collinear and A18: C, C1, A2
  are_collinear by Lm3,A12,A3,A4,A10;
  A19: A <> A1 by Th14,A1,A2;
  C2, A, A1 are_collinear by A13;
  then A20: C2 in Line(A, A1) by A19,Th13;
  B2, A, A1 are_collinear by A16;
  then A21: B2 in Line(A, A1) by A19,Th13;
A22:  A, A1, B2, C2 are_collinear by A13,A16,A19,RLTOPSP1:81;
  B, C, A is_a_triangle by A1;
  then B <> B1 by Th14,A3;
  then
A23:  B, B1, A2, C2 are_collinear by A14,A17,RLTOPSP1:81;
  C, A, B is_a_triangle by A1;
  then C <> C1 by Th14,A4;
  then
A24:  C, C1, A2, B2 are_collinear by A15,A18,RLTOPSP1:81;
  hereby
    assume
A25: (lambda / (1 - lambda)) * (mu / (1 - mu)) * (nu / (1 - nu)) = 1;
    per cases;
    suppose A26: B2 <> C2;
      take A2;
      A2, B2, C2 are_collinear by A25,Lm7,A1,A2,A3,A4,A5,A6,A7,A22,A23,A24;
      then A27: A2 in Line(B2, C2) by Th13,A26;
      Line(B2, C2) c= Line(A, A1) by A20,A21,EUCLID_4:42;
      then A2, A, A1 are_collinear by Th13,A19,A27;
      hence A, A1, A2 are_collinear;
      thus B, B1, A2 are_collinear & C, C1, A2 are_collinear by A17,A18;
    end;
    suppose A28: B2 = C2;
      take B2;
      thus A, A1, B2 are_collinear & B, B1, B2 are_collinear & C, C1, B2
      are_collinear by A28,A14,A15,A16;
    end;
  end;
  given C3 such that A29: A, A1, C3 are_collinear and A30: B, B1, C3
  are_collinear and A31: C, C1, C3 are_collinear;
  A32: C3, B2, C2 are_collinear
  proof
    per cases;
    suppose A33: B2 <> C2;
      C3, A, A1 are_collinear by A29;
      then A34: C3 in Line(A, A1) by Th13,A19;
      Line(A, A1) c= Line(B2, C2) by A20,A21,A33,EUCLID_4:43;
      hence thesis by A33,Th13,A34;
    end;
    suppose B2 = C2;
      then not C3, B2, C2 are_mutually_distinct by ZFMISC_1:def 5;
      hence thesis by EUCLID_6:20;
    end;
  end;
  C3 = A2
  proof
    assume A35: C3 <> A2;
    C, A, B is_a_triangle by A1;
    then A36: C <> C1 by Th14,A4;
    then A37: Line(C, C1) is being_line;
    B, C, A is_a_triangle by A1;
    then A38: B <> B1 by Th14,A3;
    then A39: Line(B, B1) is being_line;
    C3, B, B1 are_collinear by A30;
    then A40: C3 in Line(B, B1) by Th13,A38;
    A2, B, B1 are_collinear by A17;
    then A41: A2 in Line(B, B1) by Th13,A38;
    A2, C, C1 are_collinear by A18;
    then A42: A2 in Line(C, C1) by Th13,A36;
    C3, C, C1 are_collinear by A31;
    then C3 in Line(C, C1) by Th13,A36;
    then A43: Line(B, B1) = Line(C, C1) by A40,A35,A41,A42,A39,A37,EUCLID_4:44;
    A44: 1 - nu <> 0 by A7;
    B in Line(B, B1) by EUCLID_4:41;
    then B, C, C1 are_collinear by Th13,A36,A43;
    then the_area_of_polygon3(B, C, C1) = 0 by Th9;
    then the_area_of_polygon3(C1, B, C) = 0;
    then (1 - nu) * the_area_of_polygon3(A, B, C) + nu * the_area_of_polygon3
    (B, B, C) = 0 by Th7,A4;
    then the_area_of_polygon3(A, B, C) = 0 by A44;
    hence contradiction by Th9,A1;
  end;
  hence thesis by A32,Lm7,A1,A2,A3,A4,A5,A6,A7,A22,A23,A24;
end;
