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.