Problem
1993 AHSME Problem 27
![[asy] draw(circle((4,1),1),black+linewidth(.75)); draw((0,0)--(8,0)--(8,6)--cycle,black+linewidth(.75)); MP("A",(0,0),SW);MP("B",(8,0),SE);MP("C",(8,6),NE);MP("P",(4,1),NW); MP("8",(4,0),S);MP("6",(8,3),E);MP("10",(4,3),NW); MP("->",(5,1),E); dot((4,1)); [/asy]](/static/aops/94789fe243e37a1fab2e0a2d7f372008946209a4.png)
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\quad \text{(B) } 12\quad \text{(C) } 14\quad \text{(D) } 15\quad \text{(E) } 17
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