 TOPICS  Given two crossed ladders resting against two buildings, what is the distance between the buildings? Let the height at which they cross be and the lengths of the ladders and . The height at which touches the building is then obtained by simultaneously solving the equations   (1)   (2)

and (3)

the latter of which follows either immediately from the crossed ladders theorem or from similar triangles with , , and . Eliminating gives the equations   (4)   (5)

These quartic equations can be solved for and given known values of , , and .

There are solutions in which not only , , , , and are all integers, but so are , and . One example is . The problem can also be generalized to the situation in which the ends of the ladders are not pinned against the buildings, but propped fixed distances and away.

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## References

Gardner, M. Mathematical Circus: More Puzzles, Games, Paradoxes and Other Mathematical Entertainments from Scientific American. New York: Knopf, pp. 62-64, 1979.