Trilinear polarity

"[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.

In detail, let the line AP, BP, CP meet the sidelines BC, CA, AB at D, E, F respectively.

Let the pairs of line (BC, EF), (CA, FD), (DE, AB) intersect at X, Y, Z respectively.

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 K is a circumconic E of the triangle of reference.

Construction of a trilinear polar of a point P
Given triangle △ ABC
Cevian triangle △ DEF of △ ABC from P
Cevian lines which intersect at P
Constructed trilinear polar (line XYZ )
Construction of a trilinear pole of a line XYZ
Given trilinear polar (line XYZ )
Given triangle △ ABC
Cevian triangle △ UVW of △ ABC from XYZ
Cevian lines, which intersect at the trilinear pole P
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.