Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...

, trilinear polarity is a certain correspondence between the points in the

plane Plane most often refers to: * Aero- or airplane, a powered, fixed-wing aircraft * Plane (geometry), a flat, 2-dimensional surface * Plane (mathematics), generalizations of a geometrical plane Plane or planes may also refer to: Biology * Plane ...

of a

triangle A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...

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 Polarity may refer to: Science *Electrical polarity, direction of electrical current *Polarity (mutual inductance), the relationship between components such as transformer windings *Polarity (projective geometry), in mathematics, a duality of orde ...

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 . He is considered a reviver of projective geometry, and his work ''Traité des propriét� ...

(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 In geometry, collinearity of a set of Point (geometry), points is the property of their lying on a single Line (geometry), line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, t ...

. 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: T ...

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 :

\frac + \frac + \frac = 0.

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. is the trilinear pole of the line .

Some trilinear polars

Some of the trilinear polars are well known. *The trilinear polar of the

centroid In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the figure. The same definition extends to any object in n-d ...

of triangle is the

line at infinity In geometry and topology, the line at infinity is a projective line that is added to the affine plane in order to give closure to, and remove the exceptional cases from, the incidence properties of the resulting projective plane. The line at ...

. *The trilinear polar of the

symmedian point In geometry, symmedians are three particular lines associated with every triangle. They are constructed by taking a median of the triangle (a line connecting a vertex with the midpoint of the opposite side), and reflecting the line over the co ...

is the

Lemoine 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 ...

of triangle . *The trilinear polar of the

orthocenter The orthocenter of a triangle, usually denoted by , is the point (geometry), point where the three (possibly extended) altitude (triangle), altitudes intersect. The orthocenter lies inside the triangle if and only if the triangle is acute trian ...

is the orthic axis. *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 :

\frac + \frac + \frac = 0.

Since this passes through , :

\frac + \frac + \frac = 0.

Thus the locus of is :

\frac + \frac + \frac = 0.

This is a

circumconic In Euclidean geometry, a circumconic is a conic section that passes through the three vertices of a triangle, and an inconic is a conic section inscribed in the sides, possibly extended, of a triangle.Weisstein, Eric W. "Inconic." From MathWorld- ...

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. It can be shown that is the perspector of , namely, where and the polar triangle with respect to are perspective. The polar triangle is bounded by the tangents to at the vertices of . For example, the Trilinear polar of a point on the circumcircle must pass through its perspector, the Symmedian point X(6).

References

{{reflist

External links

*Geometrikon page
Trilinear polars
*Geometrikon page

Triangle geometry