Trilinear polarity
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 points." It was 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 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. 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 ( "x" : "y" : "z" ). Equation of the line is Since this passes through "K", Thus the locus of "P" is 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 is a circumconic of the triangle of reference.