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 $c$ and the lengths of the Ladders $a$ and $b$. The height at which $b$ touches the building $k$ is then obtained by solving

$$k^4-2ck^3+k^2(a^2-b^2)-2ck(a^2-b^2)+c^2(a^2-b^2)=0.$$

Call the horizontal distance from the top of $a$ to the crossing $u$, and the distance from the top of $b$, $v$. Call the height at which $a$ touches the building $h$. There are solutions in which $a$, $b$, $h$, $k$, $u$, and $v$ are all Integers. One is $a=119$, $b=70$, $c=30$, and $u+v=56$.

See also Ladder

References

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