The Hallway Problem

The figure below represents a corner in a hallway. The diagonal line represents a pipe that must be carried horizontally around the corner in the hallway.

What is the longest pipe that can be carried around the corner?

To answer the question, we must find the smallest of all possible pipe lengths in the diagram. Paradoxically, it is the smallest of these lengths which is in fact the longest which would fit around the corner.

Any pipe which is longer than the smallest possible in the diagram would intersect the walls of the hallway and would not pass horizontally through the corner without bending.  The smallest length pipe in the diagram only intersects at the three indicated points and because it is the smallest length, it will not intersect at any other points and will therefore just fit around the corner.

Try changing the length of the pipe by moving the point E. Change the widths of each hallway by moving points A, B and C. 

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