theorem Th57: :: EUCLID12:76

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).| = (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))) )

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).| = (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))) )