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

*x* /

*p* +

*y* /

*q* +

*z* /

*r* = 0.

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 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*.
Let *P* with trilinear coordinates ( *X* : *Y* : *Z* ) be the pole of a line passing through a fixed point *K* with trilinear coordinates ( *x*_{0} : *y*_{0} : *z*_{0} ). Equation of the line is

*x* / *X* + *y* / *Y* + *z* / *Z* =0.
Since this passes through *K*,

*x*_{0} / *X* + *y*_{0} / *Y* + *z*_{0} / *Z* =0.
Thus the locus of *P* is

*x*_{0} / *x* + *y*_{0} / *y* + *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 is a circumconic of the triangle of reference.