reserve a, b, c, d, x, y, z for Complex;
reserve r for Real;

theorem Th84:
  a <> b & b <> c & angle(a,b,c) = PI implies angle(b,c,a) = 0 &
  angle(c,a,b) = 0
proof
  assume that
A1: a <> b and
A2: b <> c and
A3: angle(a,b,c) = PI;
A4: c-b <> 0 by A2;
  set r = -Arg (a+-b);
  set A = Rotate(a+-b,r), B = Rotate(c+-b, r);
A5: Rotate(0c,r) = 0c by Th50;
A6: angle(a+-b,0c,c+-b) = angle(a+-b,b+-b,c+-b) .= angle(a,b,c) by Th70;
A7: c+-b <> a+-b by A3,Th77,COMPTRIG:5;
A8: a-b <> 0 by A1;
  then
A9: angle(a+-b,0c,c+-b) = angle(A,0c,B) by A5,A4,Th76;
  a+-b <> 0c by A1;
  then |.a+-b.| > 0 by COMPLEX1:47;
  then
A10: Im A = 0 & Re A > 0 by COMPLEX1:12,SIN_COS:31;
  then
A11: Arg A = 0c by Th19;
  then Arg(B-0c)-Arg(A-0c)>=0 by COMPTRIG:34;
  then
A12: angle(a,b,c) = Arg B by A6,A9,A11,Def4;
  angle(b,c,a) = angle(b+-b,c+-b,a+-b) by Th70
    .= angle(0c,B,A) by A5,A7,A4,Th76;
  hence angle(b,c,a) = 0 by A3,A10,A12,Lm6;
  angle(c,a,b) = angle(c+-b,a+-b,b+-b) by Th70
    .=angle(B,A,0c) by A5,A8,A7,Th76;
  hence thesis by A3,A10,A12,Lm6;
end;
