Perpendicular Bisectors example question

After a long and bloody war, the empires of Euclidia and Right Anglia have reached a tentative peace treaty over the disputed Bearing Strait. The diplomats from either side have agreed that the only fair thing to do is to divide the Strait down the middle - any point that's closer to the Euclidian airport will belong to Euclidia, and anything closer to the printing press at the Right Anglian port of High Potter will henceforth be Right Anglian territory. We got two e-mails - one from the cartographers on each side - asking almost an almost identical question: we don't trust the other guys to divide the Strait up fairly on the map, so we want to make sure they draw the dividing line exactly right - otherwise there's likely to be another long and bloody war over control of the Parallelo pipeline that runs under the Straits.

We sent the same reply to both sets of map-makers:

Dear Sirs, For this difficult and probably futile task, you're going to need a pair of compasses and a straight edge. Being cartographers, you'll probably have those handy. In your pencil cases there, look. No, don't poke each other with the pointy bits. For heaven's sake. Now, put the pointy bit in one of the cities and stretch the compass out well over halfway to the other, and draw a big circle. Without changing how wide the compass is, do the same thing from the other city. You should have two points where the circles cross. Those two points are going to be on the new border. Yes, I know it's cool. You can put the compasses back in their protective wrapping now. The last thing to do is to join them up using a ruler - extend it way beyond the crossing points and bingo! There's your new border. Assuming you haven't cheated, you should both end up with the same line.

Peace, The Maths Questions Bloggers.