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When all the points P on one line are related by an isometry to all points P^' on another, the midpoints of the segments PP^' are either distinct and collinear or coincident.

A 32-sided polygon. The regular icosidodecagon is a constructible polygon, and the regular icosidodecahedron of side length 1 has inradius r, circumradius R, and area A r = ...

A 24-sided polygon. The regular icositetragon is constructible. For side length 1, the inradius r, circumradius R, and area A are given by r = 1/2(2+sqrt(2)+sqrt(3)+sqrt(6)) ...

Two objects which touch each other are said to be incident.

A geometric figure which touches only the sides (or interior) of another figure.

A 10000-sided polygon.

One of the two pieces of a double cone (i.e., two cones placed apex to apex).

An 80-sided polygon.

The star polygon {8/3}.

One of the eight regions of space defined by the eight possible combinations of signs (+/-,+/-,+/-) for x, y, and z.