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The radical circle of the Lucas circles is the circumcircle of the Lucas tangents triangle. Its center has trilinear center function alpha_(1151)=2cosA+sinA (1) corresponding ...

A set of n magic circles is a numbering of the intersections of the n circles such that the sum over all intersections is the same constant for all circles. The above sets of ...

The Mercator series, also called the Newton-Mercator series (Havil 2003, p. 33), is the Taylor series for the natural logarithm ln(1+x) = sum_(k=1)^(infty)((-1)^(k+1))/kx^k ...

The Nechushtan graph, illustrated above, is a 10-vertex 5-chromatic graph that is unit-distance in 3 dimensions. It was used by Nechushtan (2002) in the construction of of a ...

The following table gives the number of nonadjacent vertex pairs k on graphs of n=1, 2, ... vertices. k counts 1 0, 1, 1, 1, 1, 1, 1, ... 2 0, 0, 1, 2, 2, 2, 2, ... 3 0, 0, ...

A number which is simultaneously a nonagonal number N_m and pentagonal number P_n and therefore satisfies the Diophantine equation 1/2m(7m-5)=1/2n(3n-1). (1) Completing the ...

A number which is simultaneously octagonal and heptagonal. Let O_m denote the mth octagonal number and H_n the nth heptagonal number, then a number which is both octagonal ...

A simple unlabeled graph on n vertices is called pancyclic if it contains cycles of all lengths, 3, 4, ..., n. Since a pancyclic graph must contain a cycle of length n, ...

The catacaustic of a parabola (t,t^2) opening upward is complicated for a general radiant point (x,y). However, the equations simplify substantially in the case x=infty ...

The pentagonal orthobicupola is a convex equilateral orthobicupola having regular pentagonal upper and lower faces. It is Johnson solid J_(30). The unit pentagonal ...