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 Th49:
  A,B,C is_a_triangle & A in circle(a,b,r) & B in circle(a,b,r) &
  C in circle(a,b,r)
  implies |[a,b]| = the_circumcenter(A,B,C) & r = |.the_circumcenter(A,B,C)-A.|
  proof
    assume that
A1: A,B,C is_a_triangle and
A2: A in circle(a,b,r) and
A3: B in circle(a,b,r) and
A4: C in circle(a,b,r);
    A in {p where p is Point of TOP-REAL 2: |.p - |[a,b]|.| = r}
        by A2,JGRAPH_6:def 5;
    then consider Ar be Point of TOP-REAL 2 such that
A5: Ar = A and
A6: |.Ar - |[a,b]|.| = r;
    B in {p where p is Point of TOP-REAL 2: |.p - |[a,b]|.| = r}
        by A3,JGRAPH_6:def 5;
    then consider Br be Point of TOP-REAL 2 such that
A7: Br = B and
A8: |.Br - |[a,b]|.| = r;
    C in {p where p is Point of TOP-REAL 2: |.p - |[a,b]|.| = r}
        by A4,JGRAPH_6:def 5;
    then consider Cr be Point of TOP-REAL 2 such that
A9: Cr = C and
A10: |.Cr - |[a,b]|.| = r;
A12: A,B,C are_mutually_distinct by A1,EUCLID_6:20;
A13: |.A - |[a,b]|.| = |.|[a,b]| - B.| &
    |.B - |[a,b]|.| = |.|[a,b]| - C.| &
    |.C - |[a,b]|.| = |.|[a,b]| - A.| by A5,A6,A7,A8,A9,A10,EUCLID_6:43;
    |[a,b]| in the_perpendicular_bisector(A,B) &
    |[a,b]| in the_perpendicular_bisector(B,C) &
    |[a,b]| in the_perpendicular_bisector(C,A) by A13,A12,EUCLID_6:43,Th45;
    then
A20: |[a,b]| in the_perpendicular_bisector(A,B) /\
    the_perpendicular_bisector(B,C) &
    |[a,b]| in the_perpendicular_bisector(B,C) /\
    the_perpendicular_bisector(C,A) &
    |[a,b]| in the_perpendicular_bisector(C,A) /\
    the_perpendicular_bisector(A,B) by XBOOLE_0:def 4;
    A,B,C is_a_triangle & B,C,A is_a_triangle &
    C,A,B is_a_triangle by A1,MENELAUS:15;
    then
    {|[a,b]|} = the_perpendicular_bisector(A,B) /\
    the_perpendicular_bisector(B,C) &
    {|[a,b]|} = the_perpendicular_bisector(B,C) /\
    the_perpendicular_bisector(C,A) &
    {|[a,b]|} = the_perpendicular_bisector(C,A) /\
    the_perpendicular_bisector(A,B) by A20,Th17,Th46;
    hence |[a,b]| = the_circumcenter(A,B,C) by A1,Def3;
    hence thesis by A5,A6,EUCLID_6:43;
  end;
