
theorem
  for A,B,C,P be Point of TOP-REAL 2 st A,B,C is_a_triangle &
  angle(C,B,A)<PI &
  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
  holds |.A-P.| * sin (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,C is_a_triangle and
A2: angle(C,B,A)<PI and
A3: A,B,P are_mutually_distinct and
A4: angle (P,B,A) = angle (C,B,A) /3 and
A5: angle (B,A,P) = angle (B,A,C) /3 and
A6: angle (A,P,B) < PI;
A7: angle (C,B,A)/3 + angle (B,A,C)/3 + angle (A,C,B)/3 = PI/3
    by A1,A2,Lm12;
A8: angle (A,P,B) + angle (P,B,A)+angle(B,A,P)=PI by A3,A6,EUCLID_3:47;
    |.A-P.| * sin (PI - (angle (C,B,A)/3 + angle (B,A,C)/3)) =
    |.A-B.| * sin (angle(C,B,A) /3) by A3,A4,EUCLID_6:6,A8,A5;
    hence thesis by A7,Thm1;
  end;
