In a triangle ABC of inradius r and
circumradius R (see the figure below),
the bisectors of angles A, B, and C meet the circumcircle O at A_{1}, B_{1},
and C_{1}, respectively. If the distance of A_{1}, B_{1}, and C_{1} to AC, BC, and
AC, respectively, are a_{1}, b_{1}, and c_{1}, respectively, prove that a_{1} + b_{1}
+ c_{1} = 2.(R + r).
