  A little known property of free-fall motion is the elliptic locus of the
maximum heights attained by coplanar projectiles launched from a single point
in different directions with the same initial speed. Another, less known and
perhaps somewhat surprising, property of this family consists in the said
ellipse being also the geometric locus of the critical points of the
projectiles' distances from the launch point. In the article, to gain a better
perspective on the geometry involved, we consider these loci from the
standpoint of a free-falling frame. It is shown that, in this reference frame,
the considered loci are congruent circles tangent to each other along the
horizontal and internally tangent to another, twice as big a circle, which
represents the locus of the projectiles' ranges. This simplified geometric
description is employed to recover the elliptic locus in the laboratory frame
and further explore its properties.
