Triangular coordinates

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The term triangular coordinates may refer to either of at least three related systems of coordinates in the Euclidean plane. See barycentric coordinates and ternary plot for an account of one such system, and trilinear coordinates for another. Another is that used by Stan Dolan (see References below) based on three unit vectors i, j, k at 120° angles to each other, labeled counterclockwise by convention. Any point in the plane can be (non-uniquely, since i + j + k = 0) written as

\langle x,y,z \rangle = x\mathbf{i} + y\mathbf{j} + z\mathbf{k}. \,

These are also called synergetics coordinates.

Dolan gives several lemmas:

  • \langle u,v,w \rangle = \langle x,y,z \rangle \text{ if and only if }x-u = y-v = z-w.\,
  • \left|\langle x,y,z \rangle\right| = (x + y + z)^2 - 3(xy + yz + zx). \,
  • \langle x,y,z \rangle \text{ is at }120^\circ\text{ to }\langle z,x,y \rangle
  • \langle y-z,z-x,x-y \rangle \text{ is perpendicular to } \langle x,y,z \rangle
  • Any triangle can be so positioned that its vertices are \langle a,0,0 \rangle,\ \langle 0,b,0\rangle\text{ and } \langle 0,0,c \rangle.

Dolan frequently uses the two quantities

\begin{align} S & {} = a + b + c \\ K & {} = \frac{bc + ca + ab}{a + b + c} \end{align}

to think about triangles.

Propositions about triangles that are cumbersome to prove using Cartesian coordinates can be proved in much more natural ways using this triangular coordinate system. As an example, Dolan proves Lester's theorem and related results such as basic facts about the Fermat points of a triangle.

[edit] References

  • Stan Dolan, "Man versus Computer", Mathematicsl Gazette, volume 91, number 522, November, 2007, pages 469–

[edit] External links