The Midpoint Formula

Averaging two numbers always gives you the point exactly between them on a number line. Average 2 and 10 and you get 6, sitting dead center. The midpoint formula does this twice at once, once for the x-coordinates and once for the y-coordinates, because a point in the plane is really two independent number lines glued together. For points (x1, y1) and (x2, y2), the midpoint is ((x1+x2)/2, (y1+y2)/2).

The key is that the two averages don't interact. The x-coordinate of the midpoint only cares about the two x-values; the y-coordinate only cares about the two y-values. Swap both y-values and the midpoint's x-coordinate doesn't change at all. This independence is what makes the formula fast: you're solving two one-dimensional averaging problems side by side, not one two-dimensional problem.

This sits right next to the distance formula in coordinate geometry, since both come from treating horizontal and vertical separation separately, one through the Pythagorean theorem, the other through simple averaging. Midpoint also shows up inside circle equations: the center of a circle is the midpoint of any diameter, but the midpoint of a chord that isn't a diameter pulls inside the circle rather than landing back on its edge, since a straight segment between two points on a curve cuts across the inside of that curve.