Two vertical poles, 6 m and 12 m high, stand on level ground some distance apart. A wire runs from the top of each pole to the foot of the other, and the two wires cross. At what height above the ground do they cross?

Put the poles at $x=0$ and $x=d$. The wire from (0, 6) to $(d$, 0) has equation $y=6(1-x/d)$, and the wire from (0, 0) to $(d$, 12) has equation $y=12x/d$. Setting them equal, $6-6x/d=12x/d$ gives $18x/d=6$, so $x/d=1/3$ and $y=12\times 1/3=4$ m. Notice the answer $4=(6\times 12)/(6+12)$ does not depend on how far apart the poles stand. Averaging the heights to get 9 m is the natural but wrong guess.