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The interior dimensions of a rectangular box with a lid on it are 3 feet by 4 feet by 5 feet. A post in the form of a right circular cylinder 1 foot in diameter just fits diagonally in the box, touching all six inner faces. How long is the post?

Given the three vertices of a triangle and a circle containing two vertices and the incenter of the triangle (but you are not given the center of the circle or the incenter), construct the incenter using only an (umarked) straightedge.