Harmonic Conjugate Points

Given Collinear points $W$, $X$, $Y$, and $Z$, $Y$ and $Z$ are harmonic conjugates with respect to $W$ and $X$ if

$${\vert WY\vert\over \vert YX\vert} = {\vert WZ\vert\over \vert ZX\vert}.$$

The distances between such points are said to be in Harmonic Ratio, and the Line Segment depicted above is called a Harmonic Segment.

Harmonic conjugate points are also defined for a Triangle. If $W$ and $X$ have Trilinear Coordinates $\alpha:\beta:\gamma$ and $\alpha':\beta':\gamma'$, then the Trilinear Coordinates of the harmonic conjugates are

$$\begin{eqnarray*} Y&=&\alpha+\alpha':\beta+\beta':\gamma+\gamma'\\ Z&=&\alpha-\alpha':\beta-\beta':\gamma-\gamma' \end{eqnarray*}$$

(Kimberling 1994).

See also Harmonic Range, Harmonic Ratio

References

Kimberling, C. ``Central Points and Central Lines in the Plane of a Triangle.'' Math. Mag. 67, 163-187, 1994.

Ogilvy, C. S. Excursions in Geometry. New York: Dover, pp. 13-14, 1990.

Phillips, A. W. and Fisher, I. Elements of Geometry. New York: American Book Co., 1896.

Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. New York: Viking Penguin, p. 92, 1992.