AHSME 1993 Problem 27

The sides of $\triangle ABC$ have lengths $6,8,$ and $10$. A circle with center $P$ and radius $1$ rolls around the inside of $\triangle ABC$, always remaining tangent to at least one side of the triangle. When $P$ first returns to its original position, through what distance has $P$ traveled? $\text{(A) } 10 \text{(B) } 12 \text{(C) } 14 \text{(D) } 15 \text{(E) } 17$