
theorem Thm35:
  for A,B,C,P be Point of TOP-REAL 2 st
  A,B,P are_mutually_distinct &
  angle (P,B,A) = angle (C,B,A) / 3 &
  angle (B,A,P) = angle (B,A,C) / 3 &
  angle (A,P,B) < PI &
  angle (C,B,A) / 3 + angle (B,A,C) / 3 + angle (A,C,B) / 3 = PI / 3
  holds |.A-P.| * sin (2*PI/3 + angle(A,C,B)/3) =
  |.A-B.| * sin (angle(C,B,A) / 3)
  proof
    let A,B,C,P be Point of TOP-REAL 2;
    assume that
A1: A,B,P are_mutually_distinct and
A2: angle (P,B,A) = angle (C,B,A) /3 and
A3: angle (B,A,P) = angle (B,A,C) /3 and
A4: angle (A,P,B) < PI and
A5: angle (C,B,A)/3 + angle (B,A,C)/3 + angle (A,C,B)/3 = PI/3;
    angle (A,P,B) + angle (P,B,A)+angle(B,A,P)=PI by A1,A4,EUCLID_3:47;
    hence thesis by A1,A2,A3 ,A5,EUCLID_6:6;
  end;
