Trilinear polarity
In Euclidean 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."[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.[1][2]
Definitions
Let Template:Math be a plane triangle and let Template:Mvar be any point in the plane of the triangle not lying on the sides of the triangle. Briefly, the trilinear polar of Template:Mvar is the axis of perspectivity of the cevian triangle of Template:Mvar and the triangle Template:Math.
In detail, let the line Template:Mvar meet the sidelines Template:Mvar at Template:Mvar respectively. Triangle Template:Math is the cevian triangle of Template:Mvar with reference to triangle Template:Math. Let the pairs of line Template:Math intersect at Template:Mvar respectively. By Desargues' theorem, the points Template:Mvar are collinear. The line of collinearity is the axis of perspectivity of triangle Template:Math and triangle Template:Math. The line Template:Mvar is the trilinear polar of the point Template:Mvar.[1]
The points Template:Mvar can also be obtained as the harmonic conjugates of Template:Mvar with respect to the pairs of points Template:Math respectively. Poncelet used this idea to define the concept of trilinear polars.[1]
If the line Template:Mvar is the trilinear polar of the point Template:Mvar with respect to the reference triangle Template:Math then Template:Mvar is called the trilinear pole of the line Template:Mvar with respect to the reference triangle Template:Math.
Trilinear equation
Let the trilinear coordinates of the point Template:Mvar be Template:Math. Then the trilinear equation of the trilinear polar of Template:Mvar is[3]
Construction of the trilinear pole
Let the line Template:Mvar meet the sides Template:Mvar of triangle Template:Math at Template:Mvar respectively. Let the pairs of lines Template:Math meet at Template:Mvar. Triangles Template:Math and Template:Math are in perspective and let Template:Mvar be the center of perspectivity. Template:Mvar is the trilinear pole of the line Template:Mvar.
Some trilinear polars
Some of the trilinear polars are well known.[4]
- The trilinear polar of the centroid of triangle Template:Math is the line at infinity.
- The trilinear polar of the symmedian point is the Lemoine axis of triangle Template:Math.
- The trilinear polar of the orthocenter is the orthic axis.
- Trilinear polars are not defined for points coinciding with the vertices of triangle Template:Math.
Poles of pencils of lines
Let Template:Mvar with trilinear coordinates Template:Math be the pole of a line passing through a fixed point Template:Mvar with trilinear coordinates Template:Math. Equation of the line is
Since this passes through Template:Mvar,
Thus the locus of Template:Mvar is
This is a circumconic of the triangle of reference Template:Math. Thus the locus of the poles of a pencil of lines passing through a fixed point Template:Mvar is a circumconic Template:Mvar of the triangle of reference.
It can be shown that Template:Mvar is the perspector[5] of Template:Mvar, namely, where Template:Math and the polar triangle[6] with respect to Template:Mvar are perspective. The polar triangle is bounded by the tangents to Template:Mvar at the vertices of Template:Math. 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