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 A1, B1,
and C1, respectively. If the distance of A1, B1, and C1 to AC, BC, and
AC, respectively, are a1, b1, and c1, respectively, prove that a1 + b1
+ c1 = 2.(R + r).