Question 25
Let $T_1$ be a triangle with side lengths $2011,$ $2012,$ and $2013.$ For $n \ge 1,$ if $T_n = \triangle ABC$ and $D,$ $E,$ and $F$ are the points of tangency of the incircle of $\triangle ABC$ to the sides $AB,$ $BC,$ and $AC,$ respectively, then $T_{n+1}$ is a triangle with side lengths $AD,$ $BE,$ and $CF,$ if it exists. What is the perimeter of the last triangle in the sequence $( T_n )?$