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Let A, B, and C be three circles in the plane, and let X be any circle touching B and C. Then build up a chain of circles such that Y:CAX, Z:ABY, X^':BCZ, Y^':CAX^', ...

If a polynomial P(x) is divided by (x-r), then the remainder is a constant given by P(r).

Let any finite or infinite set of points having no finite limit point be prescribed, and associate with each of its points a definite positive integer as its order. Then ...

Let G=SL(n,C). If lambda in Z^n is the highest weight of an irreducible holomorphic representation V of G, (i.e., lambda is a dominant integral weight), then the G-map ...

Each point in the convex hull of a set S in R^n is in the convex combination of n+1 or fewer points of S.

product_(k=1)^(n)(1+yq^k) = sum_(m=0)^(n)y^mq^(m(m+1)/2)[n; m]_q (1) = sum_(m=0)^(n)y^mq^(m(m+1)/2)((q)_n)/((q)_m(q)_(n-m)), (2) where [n; m]_q is a q-binomial coefficient.

Any row r and column s of a determinant being selected, if the element common to them be multiplied by its cofactor in the determinant, and every product of another element ...

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 ...

If the lines joining corresponding points of two directly similar figures are divided proportionally, then the locus of the points of the division will be a figure directly ...

A closed embedded smooth plane curve has at least four vertices, where a vertex is defined as an extremum of curvature.