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Let O and I be the circumcenter and incenter of a triangle with circumradius R and inradius r. Let d be the distance between O and I. Then d^2=R(R-2r) (Mackay 1886-1887; ...

A curve on the unit sphere S^2 is an eversion if it has no corners or cusps (but it may be self-intersecting). These properties are guaranteed by requiring that the curve's ...

Draw a triangle DeltaA_1A_2A_3, and let A_1^' be the intersection of the parallel to A_3A_1 through A_2 (the A_2-exmedian) and the parallel to A_1A_2 through A_3 (the ...

On a Lie group, exp is a map from the Lie algebra to its Lie group. If you think of the Lie algebra as the tangent space to the identity of the Lie group, exp(v) is defined ...

The exterior derivative of a function f is the one-form df=sum_(i)(partialf)/(partialx_i)dx_i (1) written in a coordinate chart (x_1,...,x_n). Thinking of a function as a ...

A variation of chess involving a change in the form of the board, the rules of play, or the pieces used. For example, the normal rules of chess can be used but with a ...

A circumconic hyperbola, which therefore passes through the orthocenter, is a rectangular hyperbola, and has center on the nine-point circle. Its circumconic parameters are ...

Consider a Lucas sequence with P>0 and Q=+/-1. A Fibonacci pseudoprime is a composite number n such that V_n=P (mod n). There exist no even Fibonacci pseudoprimes with ...

The eigenvector corresponding to the second smallest eigenvalue (i.e., the algebraic connectivity) of the Laplacian matrix of a graph G. The Fiedler vector is used in ...

A theorem due to Conway et al. (1997) which states that, if a positive definite quadratic form with integer matrix entries represents all natural numbers up to 15, then it ...