reserve n for Nat;
reserve i for Integer;
reserve r,s,t for Real;
reserve An,Bn,Cn,Dn for Point of TOP-REAL n;
reserve L1,L2 for Element of line_of_REAL n;
reserve A,B,C for Point of TOP-REAL 2;
reserve D for Point of TOP-REAL 2;
reserve a,b,c,d for Real;

theorem Th61:
  A,C,B is_a_triangle & angle(A,C,B) < PI implies
  (angle(B,A,C) - angle(C,B,A)) / 2
  = arctan (cot (angle(A,C,B) / 2) * ((|.C-B.|-|.C-A.|) / (|.C-B.| + |.C-A.|)))
  proof
    assume that
A1: A,C,B is_a_triangle and
A2: angle(A,C,B) < PI;
    set r = ((angle(B,A,C) - angle(C,B,A))/2);
A3: tan r = cot (angle(A,C,B) / 2) * ((|.C-B.|-|.C-A.|) / (|.C-B.| + |.C-A.|))
      by A1,A2,Th60;
    0 <= angle(A,C,B) by EUCLID11:2;
    then
A4: 0 < angle(A,C,B) < PI &
    A,C,B are_mutually_distinct by A1,A2,EUCLID10:30,EUCLID_6:20;
    0 < angle(B,A,C) < PI &
    A,B,C is_a_triangle by A4,EUCLID11:5,A1,MENELAUS:15;
    then -PI/2 < r < PI/2 by Th29;
    hence thesis by A3,SIN_COS9:35;
  end;
