Trilinear polarity


In Euclidean geometry, trilinear polarity is a correspondence defined using a special case of perspective triangles. It is 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, a French engineer and mathematician, who introduced the idea of the trilinear polar of a point in 1865.

Definitions

Let be a plane triangle and let be any point in the plane of the triangle not lying
on the sides of the triangle. Briefly, the trilinear polar of is the axis of perspectivity of the cevian triangle of and the triangle.
In detail, let the line meet the sidelines at respectively. Triangle is the cevian triangle of with reference to triangle. Let the pairs of line intersect at respectively. By Desargues' theorem, the points are collinear. The line of collinearity is the axis of perspectivity of triangle and triangle. The line is the trilinear polar of the point.
The points can also be obtained as the harmonic conjugates of with respect to the pairs of points respectively. Poncelet used this idea to define the concept of trilinear polars.
If the line is the trilinear polar of the point with respect to the reference triangle then is called the trilinear pole of the line with respect to the reference triangle.

Trilinear equation

Let the trilinear coordinates of the point be. Then the trilinear equation of the trilinear polar of is

Construction of the trilinear pole

The reverse construction instead proceeds by constructing the anticevian triangle of.
Let the line meet the sides of triangle at respectively. Let the pairs of lines meet at. Triangles and are in perspective and let be the center of perspectivity. is the trilinear pole of the line.

Some trilinear polars

Some of the trilinear polars are well known.
Let with trilinear coordinates be the pole of a line passing through a fixed point with trilinear coordinates. Equation of the line is
Since this passes through,
Thus the locus of is
This is a circumconic of the triangle of reference. Thus the locus of the poles of a pencil of lines passing through a fixed point is a circumconic of the triangle of reference.
It can be shown that is the perspector of, namely, where and the polar triangle with respect to are perspective. The polar triangle is bounded by the tangents to at the vertices of. For example, the Trilinear polar of a point on the circumcircle must pass through its perspector, the Symmedian point X.