theorem
  A,B,C is_a_triangle implies the_radius_of_the_circumcircle(A,B,C) > 0
  proof
    assume
A1: A,B,C is_a_triangle;
    then
A2: A,B,C are_mutually_distinct by EUCLID_6:20;
    assume
A3: the_radius_of_the_circumcircle(A,B,C) <= 0;
A4: |.the_circumcenter(A,B,C)-A.| = 0 by A1,A3,Def4;
    consider a,b,r such that
A5: A in circle(a,b,r) & B in circle(a,b,r) & C in circle(a,b,r) by A1,Th48;
A6: r = 0 by A1,A4,A5,Th49;
    circle(a,b,0) = {|[a,b]|} by EUCLID10:36;
    then A = |[a,b]| & B = |[a,b]| & C = |[a,b]|
      by A5,A6,TARSKI:def 1;
    hence contradiction by A2;
  end;
