Crossed Ladders Problem
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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