Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 3.5 AMC 10/12 Find the answer

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?

Pick one

Solution

Start by considering the triangle traced by $P$ as the circle moves around the triangle. It turns out this triangle is similar to the $6-8-10$ triangle (Proof: Realize that the slope of the line made while the circle is on $AC$ is the same as line $AC$ and that it makes a right angle when the circle switches from being on $AB$ to $BC$ ). Then, drop the perpendiculars as shown.
Since the smaller triangle is also a $6-8-10 = 3-4-5$ triangle, we can label the sides $EF,$ $CE,$ and $DF$ as $3x, 4x,$ and $5x$ respectively. Now, it is clear that $GB = DE + 1 = 4x + 1$ , so $AH = AG = 8 - GB = 7 - 4x$ since $AH$ and $AG$ are both tangent to the circle P at some point. We can apply the same logic to the other side as well to get $CI = 5 - 3x$ . Finally, since we have $HI = DF = 5x$ , we have $AC = 10 = (7 - 4x) + (5x) + (5 - 3x) = 12 - 2x$ , so $x = 1$ and $3x + 4x + 5x = \fbox{(B) 12}$
-Solution by Someonenumber011

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.