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 Th48:
  A,B,C is_a_triangle implies ex a,b,r st A in circle(a,b,r) &
  B in circle(a,b,r) & C in circle(a,b,r)
  proof
    assume A,B,C is_a_triangle;
    then consider D such that
    the_perpendicular_bisector(A,B) /\ the_perpendicular_bisector(B,C) = {D}
     and
    the_perpendicular_bisector(B,C) /\ the_perpendicular_bisector(C,A) = {D}
     and
    the_perpendicular_bisector(C,A) /\ the_perpendicular_bisector(A,B) = {D}
     and
A1: |.D-A.| = |.D-B.| and
A2: |.D-A.| = |.D-C.| and
    |.D-B.| = |.D-C.| by Th47;
    take a=D`1,b=D`2,r=|.D-A.|;
A3: D=|[a,b]| by EUCLID:53;
    now
      |.A-|[a,b]|.|=r by A3,EUCLID_6:43;
      hence A in circle(a,b,r) by TOPREAL9:43;
      |.B-|[a,b]|.|=r by A1,A3,EUCLID_6:43;
      hence B in circle(a,b,r) by TOPREAL9:43;
      |.C-|[a,b]|.|=r by A2,A3,EUCLID_6:43;
      hence C in circle(a,b,r) by TOPREAL9:43;
    end;
    hence thesis;
  end;
