In Euclidean geometry, trilinear polarity is a correspondence defined using a special case of perspective triangles. It is between the points (poles) in the plane of a triangle not lying on the sides of the triangle and lines (polars) 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
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.
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
x p + y q + z r = 0. {\displaystyle {\frac {x}{p}}+{\frac {y}{q}}+{\frac {z}{r}}=0.}
Construction of the trilinear pole
The reverse construction instead proceeds by constructing the anticevian triangle of P. 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.
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
Let P with trilinear coordinates X : Y : Z be the pole of a line passing through a fixed point K with trilinear coordinates x0 : y0 : z0. Equation of the line is
x X + y Y + z Z = 0. {\displaystyle {\frac {x}{X}}+{\frac {y}{Y}}+{\frac {z}{Z}}=0.}
Since this passes through K,
x 0 X + y 0 Y + z 0 Z = 0. {\displaystyle {\frac {x_{0}}{X}}+{\frac {y_{0}}{Y}}+{\frac {z_{0}}{Z}}=0.}
Thus the locus of P is
x 0 x + y 0 y + z 0 z = 0. {\displaystyle {\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 of E, namely, where △ABC and the polar triangle 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
External links Geometrikon page : Trilinear polars Geometrikon page : Isotomic conjugate of a line
