A circle touches the side BC of triangle ABC at P. Side AB and AC are produced to touch the circle at points Q and R respectively. The length of AQ is:
\( \frac{1}{2} \) (BC + CA + AB)
\( \frac{1}{2} \) (2BC + 2CA + 2AB)
\( \frac{1}{3} \) (BC + CA + AB)
\( \frac{1}{4} \) (BC + CA + AB)