theorem Thm35: :: EUCLID10:53

for A, B, C, P being Point of (TOP-REAL 2) st A,B,P are_mutually_distinct & angle (P,B,A) = (angle (C,B,A)) / 3 & angle (B,A,P) = (angle (B,A,C)) / 3 & angle (A,P,B) < PI & (((angle (C,B,A)) / 3) + ((angle (B,A,C)) / 3)) + ((angle (A,C,B)) / 3) = PI / 3 holds
|.(A - P).| * (sin (((2 * PI) / 3) + ((angle (A,C,B)) / 3))) = |.(A - B).| * (sin ((angle (C,B,A)) / 3))