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
( |.(A - B).| = - ((2 * r) * (sin (angle (A,C,B)))) & |.(B - C).| = - ((2 * r) * (sin (angle (B,A,C)))) & |.(C - A).| = - ((2 * r) * (sin (angle (C,B,A)))) )
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
( |.(A - B).| = - ((2 * r) * (sin (angle (A,C,B)))) & |.(B - C).| = - ((2 * r) * (sin (angle (B,A,C)))) & |.(C - A).| = - ((2 * r) * (sin (angle (C,B,A)))) )
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 ( |.(A - B).| = - ((2 * r) * (sin (angle (A,C,B)))) & |.(B - C).| = - ((2 * r) * (sin (angle (B,A,C)))) & |.(C - A).| = - ((2 * r) * (sin (angle (C,B,A)))) ) )
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: ( |.(A - B).| = - ((2 * r) * (sin (angle (A,C,B)))) & |.(B - C).| = - ((2 * r) * (sin (angle (B,A,C)))) & |.(C - A).| = - ((2 * r) * (sin (angle (C,B,A)))) )
the_diameter_of_the_circumcircle (A,B,C) = - (2 * r) by A1, A2, A3, Th56;
then ( |.(A - B).| = (- (2 * r)) * (sin (angle (A,C,B))) & |.(B - C).| = (- (2 * r)) * (sin (angle (B,A,C))) & |.(C - A).| = (- (2 * r)) * (sin (angle (C,B,A))) ) by A1, EUCLID10:50;
hence ( |.(A - B).| = - ((2 * r) * (sin (angle (A,C,B)))) & |.(B - C).| = - ((2 * r) * (sin (angle (B,A,C)))) & |.(C - A).| = - ((2 * r) * (sin (angle (C,B,A)))) ) ; :: thesis: verum