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
and
 |
(3) |
the latter of which follows either immediately from the crossed ladders theorem or from similar triangles with 
, 
, and 
. Eliminating 
gives the equations
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.