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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 ...

An ellipse intersects a circle in 0, 1, 2, 3, or 4 points. The points of intersection of a circle of center (x_0,y_0) and radius r with an ellipse of semi-major and ...

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 ...

An (infinite) line determined by two points (x_1,y_1) and (x_2,y_2) may intersect a circle of radius r and center (0, 0) in two imaginary points (left figure), a degenerate ...

Let C_1, C_2, C_3, and C_4 be four circles of general position through a point P. Let P_(ij) be the second intersection of the circles C_i and C_j. Let C_(ijk) be the circle ...

Given a triangle DeltaA_1A_2A_3, the points A_1, I, and J_1 lie on a line, where I is the incenter and J_1 is the excenter corresponding to A_1. Furthermore, the circle with ...

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, ...