reserve n for Nat,
        lambda,lambda2,mu,mu2 for Real,
        x1,x2 for Element of REAL n,
        An,Bn,Cn for Point of TOP-REAL n,
        a for Real;
 reserve Pn,PAn,PBn for Element of REAL n,
         Ln for Element of line_of_REAL n;
reserve A,B,C for Point of TOP-REAL 2;
reserve x,y,z,y1,y2 for Element of REAL 2;
reserve L,L1,L2,L3,L4 for Element of line_of_REAL 2;
reserve D,E,F for Point of TOP-REAL 2;
reserve b,c,d,r,s for Real;

theorem Th50:
  A,B,C is_a_triangle implies
  |.the_circumcenter(A,B,C)-A.| = |.the_circumcenter(A,B,C)-B.| &
  |.the_circumcenter(A,B,C)-A.| = |.the_circumcenter(A,B,C)-C.| &
  |.the_circumcenter(A,B,C)-B.| = |.the_circumcenter(A,B,C)-C.|
  proof
    assume
A1: A,B,C is_a_triangle;
    then consider D such that
A2: the_perpendicular_bisector(A,B) /\ the_perpendicular_bisector(B,C) = {D} &
    the_perpendicular_bisector(B,C) /\ the_perpendicular_bisector(C,A) = {D} &
    the_perpendicular_bisector(C,A) /\ the_perpendicular_bisector(A,B) = {D}
    and
A3: |.D-A.| = |.D-B.| & |.D-A.| = |.D-C.| & |.D-B.| = |.D-C.| by Th47;
    the_circumcenter(A,B,C) = D by A1,A2,Def3;
    hence thesis by A3;
  end;
