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In Euclidean geometry, trilinear polarity is a certain correspondence between the points in the
Let be a plane triangle and let be any point in the plane of the triangle not lying
on the sides of the triangle. Briefly, the trilinear polar of is the axis of perspectivity of the cevian triangle of and the triangle .
In detail, let the line meet the sidelines at respectively. Triangle is the cevian triangle of with reference to triangle . Let the pairs of line intersect at respectively. By
Let the line meet the sides of triangle at respectively. Let the pairs of lines meet at . Triangles and are in perspective and let be the
Let with trilinear coordinates be the pole of a line passing through a fixed point with trilinear coordinates . Equation of the line is
:
Since this passes through ,
:
Thus the locus of is
:
This is a circumconic of the triangle of reference . Thus the locus of the poles of a pencil of lines passing through a fixed point is a circumconic of the triangle of reference.
Trilinear polars
*Geometrikon page
Triangle geometry
plane
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* Planes (gen ...
of a triangle
A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, an ...
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
In geometry, collinearity of a set of points is the property of their lying on a single line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, the term has been used for aligned ...
." It was Jean-Victor Poncelet
Jean-Victor Poncelet (; 1 July 1788 – 22 December 1867) was a French engineer and mathematician who served most notably as the Commanding General of the École Polytechnique. He is considered a reviver of projective geometry, and his work ''Tr ...
(1788–1867), a French engineer and mathematician, who introduced the idea of the trilinear polar of a point in 1865.
Definitions
Let be a plane triangle and let be any point in the plane of the triangle not lying
on the sides of the triangle. Briefly, the trilinear polar of is the axis of perspectivity of the cevian triangle of and the triangle .
In detail, let the line meet the sidelines at respectively. Triangle is the cevian triangle of with reference to triangle . Let the pairs of line intersect at respectively. By Desargues' theorem
In projective geometry, Desargues's theorem, named after Girard Desargues, states:
:Two triangles are in perspective ''axially'' if and only if they are in perspective ''centrally''.
Denote the three vertices of one triangle by and , and tho ...
, the points are collinear. The line of collinearity is the axis of perspectivity of triangle and triangle . The line is the trilinear polar of the point .
The points can also be obtained as the harmonic conjugates of with respect to the pairs of points respectively. Poncelet
The poncelet (symbol p) is an obsolete unit of power, once used in France and replaced by (ch, metric horsepower). The unit was named after Jean-Victor Poncelet.François Cardarelli, ''Encyclopaedia of Scientific Units, Weights and Measures: The ...
used this idea to define the concept of trilinear polars.
If the line is the trilinear polar of the point with respect to the reference triangle then is called the trilinear pole of the line with respect to the reference triangle .
Trilinear equation
Let the trilinear coordinates of the point be . Then the trilinear equation of the trilinear polar of is :Construction of the trilinear pole
Let the line meet the sides of triangle at respectively. Let the pairs of lines meet at . Triangles and are in perspective and let be the center of perspectivity
Two figures in a plane are perspective from a point ''O'', called the center of perspectivity if the lines joining corresponding points of the figures all meet at ''O''. Dually, the figures are said to be perspective from a line if the points of i ...
. is the trilinear pole of the line .
Some trilinear polars
Some of the trilinear polars are well known. *The trilinear polar of the centroid of triangle is the line at infinity. *The trilinear polar of the symmedian point is theLemoine axis Lemoine or Le Moine is a French surname meaning "Monk". Notable people with the surname include:
* Adolphe Lemoine, known as Lemoine-Montigny (1812–1880), French comic-actor
* Anna Le Moine (born 1973), Swedish curler
* Antoine Marcel Lemoine ( ...
of triangle .
*The trilinear polar of the orthocenter is the orthic axis
In geometry, central lines are certain special straight lines that lie in the plane of a triangle. The special property that distinguishes a straight line as a central line is manifested via the equation of the line in trilinear coordinates. This s ...
.
*Trilinear polars are not defined for points coinciding with the vertices of triangle .
Poles of pencils of lines
Let with trilinear coordinates be the pole of a line passing through a fixed point with trilinear coordinates . Equation of the line is
:
Since this passes through ,
:
Thus the locus of is
:
This is a circumconic of the triangle of reference . Thus the locus of the poles of a pencil of lines passing through a fixed point is a circumconic of the triangle of reference.
References
{{reflistExternal links
*Geometrikon pageTrilinear polars
*Geometrikon page
Triangle geometry