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."^[1] 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.^[1]^[2]

Definitions

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 $P$ and the triangle $△ ABC$ .

In detail, let the line $AP, BP, CP$ meet the sidelines $BC, CA, AB$ at $D, E, F$ respectively. Triangle $△ DEF$ is the cevian triangle of $P$ with reference to triangle $△ ABC$ . Let the pairs of line $(BC, EF), (CA, FD), (DE, AB)$ intersect at $X, Y, Z$ respectively. By Desargues' theorem, the points $X, Y, Z$ are collinear. The line of collinearity is the axis of perspectivity of triangle $△ ABC$ and triangle $△ DEF$ . The line $XYZ$ is the trilinear polar of the point $P$ .^[1]

The points $X, Y, Z$ can also be obtained as the harmonic conjugates of $D, E, F$ with respect to the pairs of points $(B, C), (C, A), (A, B)$ respectively. Poncelet used this idea to define the concept of trilinear polars.^[1]

If the line $L$ is the trilinear polar of the point $P$ with respect to the reference triangle $△ ABC$ then $P$ 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 $p : q : r$ . Then the trilinear equation of the trilinear polar of $P$ is^[3]

{\frac {x}{p}}+{\frac {y}{q}}+{\frac {z}{r}}=0.

Construction of the trilinear pole

Let the line $L$ meet the sides $BC, CA, AB$ of triangle $△ ABC$ at $X, Y, Z$ respectively. Let the pairs of lines $(BY, CZ), (CZ, AX), (AX, BY)$ meet at $U, V, W$ . Triangles $△ ABC$ and $△ UVW$ are in perspective and let $P$ be the center of perspectivity. $P$ is the trilinear pole of the line $L$ .

Some trilinear polars

Some of the trilinear polars are well known.^[4]

The trilinear polar of the centroid of triangle $△ ABC$ 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

Animation illustrating the fact that the locus of the trilinear poles of a pencil of lines passing through a fixed point $K$ is a circumconic of the reference triangle.

Let $P$ with trilinear coordinates $X : Y : Z$ be the pole of a line passing through a fixed point $K$ with trilinear coordinates $x 0 : y 0 : z 0$ . Equation of the line is

{\frac {x}{X}}+{\frac {y}{Y}}+{\frac {z}{Z}}=0.

Since this passes through $K$ ,

{\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 $△ ABC$ . Thus the locus of the poles of a pencil of lines passing through a fixed point $K$ is a circumconic $E$ of the triangle of reference.

It can be shown that $K$ is the perspector^[5] of $E$ , namely, where $△ ABC$ and the polar triangle^[6] with respect to $E$ are perspective. The polar triangle is bounded by the tangents to $E$ 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).

References

^ ^a ^b ^c ^d Coxeter, H.S.M. (1993). The Real Projective Plane. Springer. pp. 102–103. ISBN 9780387978895.
^ Coxeter, H.S.M. (2003). Projective Geometry. Springer. pp. 29. ISBN 9780387406237.
^ Weisstein, Eric W. "Trilinear Polar". MathWorld—A Wolfram Web Resource. Retrieved 31 July 2012.
^ Weisstein, Eric W. "Trilinear Pole". MathWorld—A Wolfram Web Resource. Retrieved 8 August 2012.
^ Weisstein, Eric W. "Perspector". MathWorld—A Wolfram Web Resource. Retrieved 3 February 2023.
^ Weisstein, Eric W. "Polar Triangle". MathWorld—A Wolfram Web Resource. Retrieved 3 February 2023.

External links

Geometrikon page : Trilinear polars
Geometrikon page : Isotomic conjugate of a line

[Coxeter-1] Coxeter, H.S.M. (1993). The Real Projective Plane. Springer. pp. 102–103. ISBN 9780387978895.

[2] Coxeter, H.S.M. (2003). Projective Geometry. Springer. pp. 29. ISBN 9780387406237.

[3] Weisstein, Eric W. "Trilinear Polar". MathWorld—A Wolfram Web Resource. Retrieved 31 July 2012.

[4] Weisstein, Eric W. "Trilinear Pole". MathWorld—A Wolfram Web Resource. Retrieved 8 August 2012.

[5] Weisstein, Eric W. "Perspector". MathWorld—A Wolfram Web Resource. Retrieved 3 February 2023.

[6] Weisstein, Eric W. "Polar Triangle". MathWorld—A Wolfram Web Resource. Retrieved 3 February 2023.

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