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The locus of centers of ellipses inscribed in quadrilaterals
Alan Horwitz
Let R be a four-sided convex polygon in the xy plane and let M1 and M2 be the midpoints of the diagonals of R. It is well-known that if E is an ellipse inscribed in R, then the cen…
Invariant Means
Alan Horwitz
Let m(a,b) and M(a,b,c) be symmetric means. We say that M is type 1 invariant with respect to m if M(m(a,c),m(a,b),m(b,c)) = M(a,b,c) for all a, b, c > 0. If m is strict and isoton…
Illumination by Taylor Polynomials
Alan Horwitz
Let f(x) be a differentiable function on the real line R, and let P be a point not on the graph of f(x). Define the illumination index of P to be the number of distinct tangents to…
Compositions of Polynomials with Coefficients in a given Field
Alan Horwitz
Let F and K be fields of characteristic 0, with F a subset of K. Let K[x] denote the ring of polynomials with coefficients in K. For p in K[x]\F[x], deg(p) = n, let r be the highes…