let a be Real; :: thesis: for A, B, C being Point of (TOP-REAL 2)

for b, r being Real st A,B,C is_a_triangle & PI < angle (C,B,A) & angle (C,B,A) < 2 * 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); :: thesis: for b, r being Real st A,B,C is_a_triangle & PI < angle (C,B,A) & angle (C,B,A) < 2 * 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; :: thesis: ( A,B,C is_a_triangle & PI < angle (C,B,A) & angle (C,B,A) < 2 * 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: ( PI < angle (C,B,A) & angle (C,B,A) < 2 * PI ) and

A3: ( A in circle (a,b,r) & B in circle (a,b,r) & C in circle (a,b,r) ) ; :: thesis: 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, Th54, A4; :: thesis: verum

for b, r being Real st A,B,C is_a_triangle & PI < angle (C,B,A) & angle (C,B,A) < 2 * 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); :: thesis: for b, r being Real st A,B,C is_a_triangle & PI < angle (C,B,A) & angle (C,B,A) < 2 * 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; :: thesis: ( A,B,C is_a_triangle & PI < angle (C,B,A) & angle (C,B,A) < 2 * 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: ( PI < angle (C,B,A) & angle (C,B,A) < 2 * PI ) and

A3: ( A in circle (a,b,r) & B in circle (a,b,r) & C in circle (a,b,r) ) ; :: thesis: 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, Th54, A4; :: thesis: verum