Besides conic sections, there is one other kind of shape we can define using distance: perpendiculars. A perpendicular line is a line at a 90° angle to some reference line, but in Euclidean geometry, that's also the set of points equidistant from two points. However, rather than redefine the word perpendicular to be about distance instead of angle, I'll call the set of equidistant points the midset.
We've actually already seen what a midset looks like in taxicab geometry: it's a hyperbola with zero difference in distance between its foci. Here's one example:
One use for midsets is to find the point that's equidistant from each point of a triangle. To do so, we draw any two midsets; their intersection will be the same distance from all three corners. Here's a triangle and all three of its midsets:
To show the intersection is equidistant from all three points, we can draw a circle:
The intersection of a triangle's midsets is the circumcenter. Pleasingly, the algorithm to find it is exactly the same in taxicab geometry as in Euclidean geometry, though with the distinction that we're using midsets instead of 90° perpendicular lines.
You may recall from the discussion of hyperbolas, however, that there's a special case to check, namely when the horizontal spacing between two points equals the vertical spacing and the midset's ends flare out into areas:
Here's a triangle with a side at 45° to the axes:
One side's midset passes through the area of another side's midset. The circumcenter is now a line segment instead of a point. This happens because, with one side at 45°, two corners of the triangle fall on one side of the circumcircle and we can slide the center of the circumcircle along the intersection keeps and still keep both corners on the circumcircle.
Note that it still doesn't matter which two midsets we draw, the intersection of any two midsets are the same:
What if two sides are at 45°?
The intersection of two midsets is now a ray, in the above case, extending left from the center of the triangle's hypotenuse. This makes sense, since a triangle with two sides at 45° to the axes is one corner of a taxicab circle, no matter how big the circle is.
The same is true—but harder to see—with the midsets of the two 45° sides, which produces a one-dimensional intersection of two areas: