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A prime circle of order 2n is a free circular permutation of the numbers from 1 to 2n with adjacent pairs summing to a prime. The number of prime circles for n=1, 2, ..., are ...

A circle packing is an arrangement of circles inside a given boundary such that no two overlap and some (or all) of them are mutually tangent. The generalization to spheres ...

Draw three circles in the plane, none of which lies completely inside another, and the common external tangent lines for each pair. Then points of intersection of the three ...

Let a cone of opening parameter c and vertex at (0,0,0) intersect a sphere of radius r centered at (x_0,y_0,z_0), with the cone oriented such that its axis does not pass ...

Steiner gave and Droz-Farny (1901) proved that if equal circles are drawn about the vertices of a triangle (dashed circles in the above figure), they cut the lines joining ...

The second Brocard circle is the circle having center at the circumcenter O of the reference triangle and radius R_B = sqrt(1-4sin^2omega)R (1) = (2) where R is the ...

The inner Soddy circle is the circle tangent to each of the three mutually tangent circles centered at the vertices of a reference triangle. It has circle function ...

There are two nonintersecting circles that are tangent to all three Lucas circles. (These are therefore the Soddy circles of the Lucas central triangle.) The inner one, ...

The inverse curve of the circle with parametric equations x = acost (1) y = asint (2) with respect to an inversion circle with center (x,y) and radius R is given by x_i = ...

The radial curve of a unit circle from a radial point (x,y) and parametric equations x = cost (1) y = sint (2) is another circle with parametric equations x_r = x-cost (3) ...