Proof without words that the three utilities problem has no solution by CMG Lee. The leftmost house is first deleted. The lines connecting the utilities with the remaining houses divide the plane into three regions, shaded yellow, red and blue. Whichever region the deleted house is placed into, one of the utilities is outside the region. The Jordan curve theorem states that a line connecting them must intersect one of the existing lines.
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Shade regions with the colour of the utility outside it.
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Short title
3 utilities problem proof
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Proof without words that the three utilities problem has no solution by CMG Lee. The leftmost house is first deleted. The lines connecting the utilities with the remaining houses divide the plane into three regions, shaded yellow, red and blue. Whichever region the deleted house is placed into, one of the utilities is outside the region. The Jordan curve theorem states that a line connecting them must intersect one of the existing lines.
