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).| / (sin (angle (A,C,B))) = - (2 * r) & |.(B - C).| / (sin (angle (B,A,C))) = - (2 * r) & |.(C - A).| / (sin (angle (C,B,A))) = - (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
( |.(A - B).| / (sin (angle (A,C,B))) = - (2 * r) & |.(B - C).| / (sin (angle (B,A,C))) = - (2 * r) & |.(C - A).| / (sin (angle (C,B,A))) = - (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 ( |.(A - B).| / (sin (angle (A,C,B))) = - (2 * r) & |.(B - C).| / (sin (angle (B,A,C))) = - (2 * r) & |.(C - A).| / (sin (angle (C,B,A))) = - (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: ( |.(A - B).| / (sin (angle (A,C,B))) = - (2 * r) & |.(B - C).| / (sin (angle (B,A,C))) = - (2 * r) & |.(C - A).| / (sin (angle (C,B,A))) = - (2 * r) )
A,B,C are_mutually_distinct by A1, EUCLID_6:20;
then A4: ( C,B,A are_mutually_distinct & B,A,C are_mutually_distinct ) ;
( PI + ((2 * PI) * 0) < angle (C,B,A) & angle (C,B,A) < (2 * PI) + ((2 * PI) * 0) ) by A2;
then A5: sin (angle (C,B,A)) < 0 by SIN_COS6:12;
( PI + ((2 * PI) * 0) < angle (B,A,C) & angle (B,A,C) < (2 * PI) + ((2 * PI) * 0) ) by A2, A4, EUCLID11:8, EUCLID11:2;
then A6: sin (angle (B,A,C)) < 0 by SIN_COS6:12;
( PI + ((2 * PI) * 0) < angle (A,C,B) & angle (A,C,B) < (2 * PI) + ((2 * PI) * 0) ) by A2, A4, EUCLID11:2, EUCLID11:8;
then A7: sin (angle (A,C,B)) < 0 by SIN_COS6:12;
( |.(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, A2, A3, Th58;
then ( |.(A - B).| / (sin (angle (A,C,B))) = - (((2 * r) * (sin (angle (A,C,B)))) / (sin (angle (A,C,B)))) & |.(B - C).| / (sin (angle (B,A,C))) = - (((2 * r) * (sin (angle (B,A,C)))) / (sin (angle (B,A,C)))) & |.(C - A).| / (sin (angle (C,B,A))) = - (((2 * r) * (sin (angle (C,B,A)))) / (sin (angle (C,B,A)))) ) ;
hence ( |.(A - B).| / (sin (angle (A,C,B))) = - (2 * r) & |.(B - C).| / (sin (angle (B,A,C))) = - (2 * r) & |.(C - A).| / (sin (angle (C,B,A))) = - (2 * r) ) by A5, A6, A7, XCMPLX_1:89; :: thesis: verum