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;
