It is well known that if $\Phi _n (x)$ is the $n$th cyclotomic polynomial, then there is a factorization $x^n - 1 = \prod \Phi _d (x)$, where the product is taken over the divisors $d$ of $n$. Thus, one can obtain, by Möbius inversion, a product formula for each $\Phi _n (x)$ in terms of the various factors $x^d - 1$. The purpose of this note is two-fold. First, we show that the above factorization implies a similar factorization for the minimal polynomials of the algebraic numbers $\cos (2\pi / n)$, where $n$ is a positive integer. Secondly, we give an explicit formula for the minimal polynomials of $\cos (2\pi / p)$, where $p$ is prime.

A congruent-incircle cevian of a triangle subdivides the triangle into two triangles with congruent incircles. We give two Euclidean constructions of such cevians, and also construct Heronian triangles for which the three pairs of congruent-incircle subtriangles are all Heronian.

Let $X$ and $Y$ be two Hilbert spaces or Banach spaces, and let $T \colon X \rightarrow Y$ be a bounded linear operator with closed range. Let $\tilde{T} = T + \delta T$ with $\|\delta T\| \|T^{\dagger}\| < 1$. We give some equivalent conditions for the generalized inverse of $\tilde{T}$ to have the simplest expression $\tilde{T}^{\dagger} = (I + T^{\dagger}\delta T)^{-1} T^{\dagger} = T^{\dagger} (I + \delta T T^{\dagger})^{-1}$.

A locally multiplicative convex algebra $E$ is commutative modulo the radical $R(E)$ if and only if the left joint spectrum of a finite set of elements is contained in the right joint spectrum of these elements.