 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
  (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) implies (A1, B1, C1 are_collinear iff
  (lambda / (1 - lambda)) * (mu / (1 - mu)) * (nu / (1 - nu)) = -1)
proof
  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 & mu <> 1 & nu <> 1;
  A6: the_area_of_polygon3(A, B, C) <> 0 by Th9,A1;
  set q = ((1 - lambda) * (1 - mu) * (1 - nu));
  A7: (1 - lambda) <> 0 & (1 - mu) <> 0 & (1 - nu) <> 0 by A5;
  A1, B1, C1 are_collinear iff the_area_of_polygon3(A1, B1, C1) = 0 by Th9;
  then  A1, B1, C1 are_collinear iff
  (q + lambda * mu * nu) * the_area_of_polygon3(A, B, C) = 0
  by Th17,A2,A3,A4;
  then A1, B1, C1 are_collinear iff 1 * q + (lambda * mu * nu) = 0 by A6;
  then A1, B1, C1 are_collinear iff q * (1 + (lambda * mu * nu) / q) = 0
  by A7,XCMPLX_1:235;
  then A1, B1, C1 are_collinear iff 1 + (lambda * mu * nu) / q = 0 by A7;
  then A1, B1, C1 are_collinear iff - 1 = (lambda * (mu * nu)) /
  ((1 - lambda) * ((1 - mu) * (1 - nu)));
  then A1, B1, C1 are_collinear iff - 1 =
  ((lambda / (1 - lambda)) * (mu * nu))
  / ((1 - mu) * (1 - nu)) by XCMPLX_1:83;
  then A1, B1, C1 are_collinear iff - 1 = (lambda / (1 - lambda)) * ((mu * nu)
  / ((1 - mu) * (1 - nu))) by XCMPLX_1:74;
  then A1, B1, C1 are_collinear iff - 1 = (lambda / (1 - lambda)) * (((mu /
  (1 - mu)) * nu) / (1 - nu)) by XCMPLX_1:83;
  then A1, B1, C1 are_collinear iff - 1 = (lambda / (1 - lambda)) * ((mu /
  (1 - mu)) * (nu / (1 - nu))) by XCMPLX_1:74;
  hence thesis;
end;
