Midpoint3 2008-11-04 17:36:11 2008-11-05 14:54:07 Definition midpoint % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions \usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \theoremstyle{definition} \newtheorem*{thmplain}{Theorem} \PMlinkescapeword{degree} The concept of \PMlinkname{midpoint of line segment}{Midpoint} is a special case of the midpoint of a curve or arbitrary figure in $\mathbb{R}^2$ or $\mathbb{R}^3$. A point $T$ is a {\em midpoint} of the figure $f$, if for each point $A$ of $f$ there is a point $B$ of $f$ such that $T$ is the midpoint of the line segment $AB$.\, One says also that $f$ is symmetric about the point $T$.\\ Given the equation of a curve in $\mathbb{R}^2$ or of a surface $f$ in $\mathbb{R}^3$, one can, if \PMlinkescapetext{necessary}, take a new point $T$ for the origin by using the linear substitutions of the form $$x := x'\!+\!a, \quad y := y'\!+\!b \quad \mbox{etc.}$$ Thus one may test whether the origin is the midpoint of $f$ by checking whether $f$ always contains along with any point\, $(x,\,y,\,z)$\, also the point\, $(-x,\,-y,\,-z)$.\\ It is easily verified the \textbf{Theorem.}\, If the origin is the midpoint of a quadratic curve or a quadratic surface, then its equation has no \PMlinkname{terms of degree}{BasicPolynomial} 1. Similarly one can verify the generalisation, that if the origin is the midpoint of an algebraic curve or surface of degree $n$, the equation has no terms of degree $n\!-\!1$,\, $n\!-\!3$\, and so on.\\ \textbf{Note.}\, Some curves and surfaces have infinitely many midpoints (see \PMlinkname{quadratic surfaces}{QuadraticSurfaces}). \begin{thebibliography}{8} \bibitem{IF}{\sc Felix Iversen}: {\em Analyyttisen geometrian oppikirja}. Tiedekirjasto Nr. 19.\, Second edition.\, Kustannusosakeyhti\"o Otava, Helsinki (1963). \end{thebibliography}