Hexagon Parallel to Medial Triangle

Choose a point on a side of a triangle. Starting with this point, draw a line parallel to one of the sides of the medial triangle. Find the point of intersection of that line with another side of the triangle and, from there continue the process. The process generates a broken line. Prove that the sixth leg of this line terminates at the starting point. In other words, the process leads to a (closed) hexagon with sides parallel to the sides of the reference triangle.

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