
06560937
j
2016c.00767
Matsuura, Ryota
Sword, Sarah
Illuminating coordinate geometry with algebraic symmetry.
Math. Teach. (Reston) 108, No. 6, 470473 (2015).
2015
National Council of Teachers of Mathematics (NCTM), Reston, VA
EN
G74
H24
symmetric polynomials
coordinate geometry
analytic geometry
upper secondary
triangles
proofs
centroid
center of mass
medians
equations of straight lines
parametric equations
generalization
tetrahedra
concurrency
reasoning
http://www.nctm.org/Publications/mathematicsteacher/2015/Vol108/Issue6/IlluminatingCoordinateGeometrywithAlgebraicSymmetry/
From the text: A symmetric polynomial is a polynomial in one or more variables in which swapping any pair of variables leaves the polynomial unchanged. For example, $f(x,y,z)=xy+xz+yz$ is a symmetric polynomial. If we interchange the variables $x$ and $y$, we obtain $yx+yz+xz$, which is the same as $f(x,y,z)$; likewise, swapping $x$ and $z$ (or $y$ and $z$) returns the original polynomial. These polynomials arise in many areas of mathematics, including Galois theory and combinatorics, but they are rarely taught in a high school curriculum. In this article, we describe an application of symmetric polynomials to a familiar problem in coordinate geometry, thus introducing this powerful tool in a context that is accessible to high school students.