# Moving Legs of an Equilateral Triangle

Geometry Level 5

Let $$ABC$$ be an equilateral triangle in the plane with side length of $$1$$ with vertices labeled in counterclockwise order. Let point $$A'$$ coincide with $$A$$, $$B'$$ coincide with $$B$$, and $$C'$$ coincide with $$C$$. Suppose the leg $$AB'$$ rotates about $$A$$ counterclockwise, $$CA'$$ rotates about $$C$$ counterclockwise, and $$BC'$$ rotates about $$B$$ counterclockwise all at the same rate until the three legs intersect at one point. Each leg intersects another leg at a point. These $$3$$ intersection points trace out paths as the legs rotate. Find the total length of these paths. Note: The diagram is not completely accurate or to scale.

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