The infographic below shows a triangle
ABC with the excenters E_{1}, E_{2}, and E_{3}.
O_{1}, O_{2}, and O_{3} are
the circumcenters of triangles E_{1}BC, E_{2}AC,
and E_{3}AB, respectively. If S_{1}, S_{2}, and
S_{3} are the areas of triangle E_{1}E_{2}E_{3},
hexagon AO_{3}BO_{1}CO_{2},
and triangle O_{1}O_{2}O_{3},
respectively, prove that
S_{1} = 2.S_{2} = 4.S_{3}
