Point D lies inside the triangle A B C. If A_{1}, B_{1}, and C_{1} are the second intersection points of the lines A D, B D, and C D with the circles circumscribed about \triangle B D C, \triangle C D A, and \triangle A D B, prove that
\frac{A D}{A A_{1}}+\frac{B D}{B B_{1}}+\frac{C D}{C C_{1}}=1