let A, B, C be Point of (TOP-REAL 2); :: thesis: ( A,B,C is_a_triangle implies the_diameter_of_the_circumcircle (A,B,C) = - (|.(C - A).| / (sin (angle (A,B,C)))) )
assume A,B,C is_a_triangle ; :: thesis: the_diameter_of_the_circumcircle (A,B,C) = - (|.(C - A).| / (sin (angle (A,B,C))))
then the_diameter_of_the_circumcircle (A,B,C) = |.(C - A).| / (sin (angle (C,B,A))) by Thm29
.= |.(C - A).| / (- (sin (angle (A,B,C)))) by EUCLID_6:2
.= - (|.(C - A).| / (sin (angle (A,B,C)))) by XCMPLX_1:188 ;
hence the_diameter_of_the_circumcircle (A,B,C) = - (|.(C - A).| / (sin (angle (A,B,C)))) ; :: thesis: verum