let a be Real; for A, B, C being Point of (TOP-REAL 2)
for b, r being Real st A,B,C is_a_triangle & 0 < angle (C,B,A) & angle (C,B,A) < PI & A in circle (a,b,r) & B in circle (a,b,r) & C in circle (a,b,r) holds
the_diameter_of_the_circumcircle (A,B,C) = 2 * r
let A, B, C be Point of (TOP-REAL 2); for b, r being Real st A,B,C is_a_triangle & 0 < angle (C,B,A) & angle (C,B,A) < PI & A in circle (a,b,r) & B in circle (a,b,r) & C in circle (a,b,r) holds
the_diameter_of_the_circumcircle (A,B,C) = 2 * r
let b, r be Real; ( A,B,C is_a_triangle & 0 < angle (C,B,A) & angle (C,B,A) < PI & A in circle (a,b,r) & B in circle (a,b,r) & C in circle (a,b,r) implies the_diameter_of_the_circumcircle (A,B,C) = 2 * r )
assume that
A1:
A,B,C is_a_triangle
and
A2:
( 0 < angle (C,B,A) & angle (C,B,A) < PI )
and
A3:
( A in circle (a,b,r) & B in circle (a,b,r) & C in circle (a,b,r) )
; the_diameter_of_the_circumcircle (A,B,C) = 2 * r
A4:
( the_diameter_of_the_circumcircle (A,B,C) = 2 * r or the_diameter_of_the_circumcircle (A,B,C) = - (2 * r) )
by A1, A3, Th52;
r > 0
by A1, A3, EUCLID10:37;
hence
the_diameter_of_the_circumcircle (A,B,C) = 2 * r
by A1, A2, Th53, A4; verum