theorem :: EUCLID12:78

for a being 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

( |.(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 )

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

( |.(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 )