The figure shows a scalene triangle ABC with
the circumcircle of center O and angle B = 60 degrees. The Euler line OH intersects
arc AB, AB, BC, and arc BC at C_{2}, C_{1}, A_{1}, and A_{2},
respectively. Prove that (1) OH = C_{1}C_{2}  A_{1}A_{2};
(2) OH = AC_{1}  CA_{1}; (3) OH = AB  BC; (4) OC_{1}
= HA_{1}.
