In the figure below, equilateral triangles ABC_{1} and
A_{1}BC are drawn on the sides of a triangle ABC.
If B_{1}, B_{2}, B_{3}, and B_{4} are the midpoints of AC,
AC_{1},
A_{1}C, and A_{1}C_{1}, respectively,
prove that (1) triangles B_{1}B_{2}B_{4}
and B_{1}B_{3}B_{4} are equilateral;
(2) B_{1}B_{2}B_{4}B_{3}
is a rhombus; (3) angle B_{2}B_{1}B_{3} = angle B_{2}B_{4}B_{3} = 120
degrees; (4) B_{2}B_{3} and B_{1}B_{4}
are perpendicular.
