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Given three mutually tangent circles, there exist exactly two nonintersecting circles which are tangent circles to all three original circles. These are called the inner and ...

The number of regions into which space can be divided by n mutually intersecting spheres is N=1/3n(n^2-3n+8), giving 2, 4, 8, 16, 30, 52, 84, ... (OEIS A046127) for n=1, 2, ...

Dissect a triangle into smaller triangles, such that all have full edge contact with their neighbors. Label the corners 1, 2, and 3. Label all vertices with 1, 2, or 3, with ...

The equation of the curve of intersection of a torus with a plane perpendicular to both the midplane of the torus and to the plane x=0. (The general intersection of a torus ...

A figure formed by taking a series of steps of length 1, 2, ..., n, with an angle theta turn after each step. The symbol for a spirolateral is ^(a_1,...,a_k)n_theta, where ...

Gardner showed how to dissect a square into eight and nine acute scalene triangles. W. Gosper discovered a dissection of a unit square into 10 acute isosceles triangles, ...

The square gyrobicupola is a convex equilateral solid that is Johnson solid J_(29). The unit square gyrobicupola has volume V=2+4/3sqrt(2) (1) and Dehn invariant D = 24<3>_2 ...

The square orthobicupola is a convex equilateral solid that is Johnson solid J_(28). The unit square orthobicupola has volume V=2+4/3sqrt(2) (1) and Dehn invariant D = ...

The Stammler circle is the circumcircle of the Stammler triangle. Unexpectedly, it has center at the circumcenter O (X_3) of the reference triangle and radius 2R (Ehrmann and ...

The Stammler hyperbola of a triangle is the Feuerbach hyperbola of its tangential triangle, and its center is the focus of the Kiepert parabola, which is Kimberling center ...