Three cottages stand at P(-4, 0), Q(0, 8) and R(4, 0) on a plan marked in metres. A well is to be dug at the one spot that is the same distance from all three cottages.
How far will the well be from each cottage?

P and R are mirror images in the $y$-axis, so the well must lie on the $y$-axis, say at (0, k). Setting its distance to P equal to its distance to Q gives $\sqrt{16+k^2}=|8-k|$, so $16+k^2=64-16k+k^2$, which leaves $16k=48$ and $k=3$. The well is at (0, 3), and $\sqrt{(-4-0{)}^2+(0-3{)}^2}=\sqrt{16+9}=5$ m to P, the same 5 m to R, and |8 - 3| = 5 m to Q. The value 4 m is only half of PR, and $4\sqrt{5}$ is the side PQ.