Trilinear polar

In 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 lines." It was Poncelet (1788–1867), a French engineer and mathematician, who introduced the idea of the trilinear polar of a point in 1865.

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.

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.

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.

Let the trilinear coordinates of the point P be (p : q : r). Then the trilinear equation of the trilinear polar of P is

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.

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