Trilinear polarity

In Euclidean geometry, trilinear polarity is a certain correspondence between the points in the plane of a triangle not lying on the sides of the triangle and lines in the plane of the triangle not passing through the  vertices of the triangle. "Although it is called a polarity, it is not really a polarity at all, for poles of concurrent lines are not collinear points." It was Jean-Victor Poncelet (1788–1867), a French engineer and mathematician, who introduced the idea of the trilinear polar of a point in 1865.

Definitions
Trilinear Polar.svg $△ABC$ of $△DEF$ from $P$}}

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Let $△ABC$ be a plane triangle and let $P$ be any point in the plane of the triangle not lying on the sides of the triangle. Briefly, the trilinear polar of $P$ is the axis of perspectivity of the cevian triangle of $XYZ$ and the triangle $△ABC$.

In detail, let the line $P$ meet the sidelines $P$ at $P$ respectively. Triangle $△ABC$ is the cevian triangle of $AP, BP, CP$ with reference to triangle $△DEF$. Let the pairs of line $△ABC$ intersect at $BC, CA, AB$ respectively. By Desargues' theorem, the points $D, E, F$ are collinear. The line of collinearity is the axis of perspectivity of triangle $(BC, EF), (CA, FD), (DE, AB)$ and triangle $△ABC$. The line $P$ is the trilinear polar of the point $X, Y, Z$.

The points $X, Y, Z$ can also be obtained as the harmonic conjugates of $XYZ$ with respect to the pairs of points $△DEF$ respectively. Poncelet used this idea to define the concept of trilinear polars.

If the line $P$ is the trilinear polar of the point $X, Y, Z$ with respect to the reference triangle $(B, C), (C, A), (A, B)$ then $D, E, F$ is called the trilinear pole of the line $L$ with respect to the reference triangle $△ABC$.

Trilinear equation
Let the trilinear coordinates of the point  $P$ be $△ABC$. Then the trilinear equation of the trilinear polar of $P$ is
 * $$\frac{x}{p} + \frac{y}{q} + \frac{z}{r} = 0.$$

Construction of the trilinear pole
[[File:Trilinear Pole.svg|thumb|338px|right|Construction of a trilinear pole of a line $L$

{{legend|#deb9a0|Given triangle $p : q : r$}} {{legend|#82ecfa|Cevian triangle $△ABC$ of $△UVW$ from $P$}} ]] Let the line $P$ meet the sides $XYZ$ of triangle $△ABC$ at $XYZ$ respectively. Let the pairs of lines $△ABC$ meet at $XYZ$. Triangles $(BY, CZ), (CZ, AX), (AX, BY)$ and $△ABC$ are in perspective and let $P$ be the center of perspectivity. $L$ is the trilinear pole of the line $BC, CA, AB$.

Some trilinear polars
Some of the trilinear polars are well known.


 * The trilinear polar of the centroid of triangle $△UVW$ is the line at infinity.
 * The trilinear polar of the symmedian point is the Lemoine axis of triangle $△ABC$.
 * The trilinear polar of the orthocenter is the orthic axis.
 * Trilinear polars are not defined for points coinciding with the vertices of triangle $△ABC$.

Poles of pencils of lines


Let $X, Y, Z$ with trilinear coordinates $△ABC$ be the pole of a line passing through a fixed point $U, V, W$ with trilinear coordinates $X : Y : Z$. Equation of the line is
 * $$\frac{x}{X} + \frac{y}{Y} + \frac{z}{Z} = 0.$$

Since this passes through $P$,
 * $$\frac{x_0}{X} + \frac{y_0}{Y} + \frac{z_0}{Z} = 0.$$

Thus the locus of $P$ is
 * $$\frac{x_0}{x} + \frac{y_0}{y} + \frac{z_0}{z} = 0.$$

This is a circumconic of the triangle of reference $x0 : y0 : z0$. Thus the locus of the poles of a pencil of lines passing through a fixed point $L$ is a circumconic $K$ of the triangle of reference.

It can be shown that $P$ is the perspector of $K$, namely, where $△ABC$ and the polar triangle with respect to $K$ are perspective. The polar triangle is bounded by the tangents to $P$ at the vertices of $△ABC$. For example, the Trilinear polar of a point on the circumcircle must pass through its perspector, the Symmedian point X(6).