An application of Bezout’s theorem

We’re buying a house today.

It turns out that although we’re moving about ten miles, both of our commutes stay about the same length. Both before and after the move, I’m going about 35 minutes and she’s going about 15.

Say my commute length is x and my wife’s commute length is y. You can imagine drawing a circle of radius x around my work and a circle of radius y around hers. (These are “circles” where the metric is driving time, not driving distance or distance as the crow flies.)  These circles intersect in two places, like circles do. (Bezout’s theorem says four, but two of those are at infinity, and there’s no way I’m living there.) We used to live at one of the intersections of these circles; we’re mvoing to the other one.

In the interest of security by obscurity, no map will be provided. Also, drawing a map with isochrones or with distances calculated along the road network isn’t really how I want to spend my morning.