Trilinear polarity

[[File:Trilinear Polar.svg|thumb|338px|right|Construction of a trilinear polar of a point {{mvar|P}}

{{legend|#deb9a0|Given triangle {{math|△ABC}}}}

{{legend|#5552fa|Cevian triangle {{math|△DEF}} of {{math|△ABC}} from {{mvar|P}}}}

{{legend-line|solid lime|Cevian lines which intersect at {{mvar|P}}}}

{{legend-line|solid red|Constructed trilinear polar (line {{mvar|XYZ}})}}]]

Let {{math|△ABC}} be a plane triangle and let {{mvar|P}} be any point in the plane of the triangle not lying

on the sides of the triangle. Briefly, the trilinear polar of {{mvar|P}} is the axis of perspectivity of the cevian triangle of {{mvar|P}} and the triangle {{math|△ABC}}.

In detail, let the line {{mvar|AP, BP, CP}} meet the sidelines {{mvar|BC, CA, AB}} at {{mvar|D, E, F}} respectively. Triangle {{math|△DEF}} is the cevian triangle of {{mvar|P}} with reference to triangle {{math|△ABC}}. Let the pairs of line {{math|(BC, EF), (CA, FD), (DE, AB)}} intersect at {{mvar|X, Y, Z}} respectively. By Desargues' theorem, the points {{mvar|X, Y, Z}} are collinear. The line of collinearity is the axis of perspectivity of triangle {{math|△ABC}} and triangle {{math|△DEF}}. The line {{mvar|XYZ}} is the trilinear polar of the point {{mvar|P}}.

The points {{mvar|X, Y, Z}} can also be obtained as the harmonic conjugates of {{mvar|D, E, F}} with respect to the pairs of points {{math|(B, C), (C, A), (A, B)}} respectively. Poncelet used this idea to define the concept of trilinear polars.

If the line {{mvar|L}} is the trilinear polar of the point {{mvar|P}} with respect to the reference triangle {{math|△ABC}} then {{mvar|P}} is called the trilinear pole of the line {{mvar|L}} with respect to the reference triangle {{math|△ABC}}.

Let the trilinear coordinates of the point {{mvar|P}} be {{math|p : q : r}}. Then the trilinear equation of the trilinear polar of {{mvar|P}} is{{cite web|last=Weisstein|first=Eric W.|title=Trilinear Polar|url=https://mathworld.wolfram.com/TrilinearPolar.html|publisher=MathWorld—A Wolfram Web Resource|accessdate=31 July 2012}}

:$\backslash frac\{x\}\{p\}\; +\; \backslash frac\{y\}\{q\}\; +\; \backslash frac\{z\}\{r\}\; =\; 0.$

[[File:Trilinear Pole.svg|thumb|338px|right|Construction of a trilinear pole of a line {{mvar|XYZ}}

{{legend-line|solid red|Given trilinear polar (line {{mvar|XYZ}})}}

{{legend|#deb9a0|Given triangle {{math|△ABC}}}}

{{legend|#82ecfa|Cevian triangle {{math|△UVW}} of {{math|△ABC}} from {{mvar|XYZ}}}}

{{legend-line|solid lime|Cevian lines, which intersect at the trilinear pole {{mvar|P}}}}]]

Let the line {{mvar|L}} meet the sides {{mvar|BC, CA, AB}} of triangle {{math|△ABC}} at {{mvar|X, Y, Z}} respectively. Let the pairs of lines {{math|(BY, CZ), (CZ, AX), (AX, BY)}} meet at {{mvar|U, V, W}}. Triangles {{math|△ABC}} and {{math|△UVW}} are in perspective and let {{mvar|P}} be the center of perspectivity. {{mvar|P}} is the trilinear pole of the line {{mvar|L}}.

Some of the trilinear polars are well known.{{cite web|last=Weisstein|first=Eric W.|title=Trilinear Pole|url=https://mathworld.wolfram.com/TrilinearPole.html|publisher=MathWorld—A Wolfram Web Resource.|accessdate=8 August 2012}}

• The trilinear polar of the centroid of triangle {{math|△ABC}} is the line at infinity.
• The trilinear polar of the symmedian point is the Lemoine axis of triangle {{math|△ABC}}.
• The trilinear polar of the orthocenter is the orthic axis.
• Trilinear polars are not defined for points coinciding with the vertices of triangle {{math|△ABC}}.

File:Trilinear poles of a pencil of lines.gif

Let {{mvar|P}} with trilinear coordinates {{math|X : Y : Z}} be the pole of a line passing through a fixed point {{mvar|K}} with trilinear coordinates {{math|x{{sub|0}} : y{{sub|0}} : z{{sub|0}}}}. Equation of the line is

:$\backslash frac\{x\}\{X\}\; +\; \backslash frac\{y\}\{Y\}\; +\; \backslash frac\{z\}\{Z\}\; =\; 0.$

Since this passes through {{mvar|K}},

:$\backslash frac\{x\_0\}\{X\}\; +\; \backslash frac\{y\_0\}\{Y\}\; +\; \backslash frac\{z\_0\}\{Z\}\; =\; 0.$

Thus the locus of {{mvar|P}} is

:$\backslash frac\{x\_0\}\{x\}\; +\; \backslash frac\{y\_0\}\{y\}\; +\; \backslash frac\{z\_0\}\{z\}\; =\; 0.$

This is a circumconic of the triangle of reference {{math|△ABC}}. Thus the locus of the poles of a pencil of lines passing through a fixed point {{mvar|K}} is a circumconic {{mvar|E}} of the triangle of reference.

It can be shown that {{mvar|K}} is the perspector{{cite web|last=Weisstein|first=Eric W.|title=Perspector|url=https://mathworld.wolfram.com/Perspector.html|publisher=MathWorld—A Wolfram Web Resource.|accessdate=3 February 2023}} of {{mvar|E}}, namely, where {{math|△ABC}} and the polar triangle{{cite web|last=Weisstein|first=Eric W.|title=Polar Triangle|url=https://mathworld.wolfram.com/PolarTriangle.html|publisher=MathWorld—A Wolfram Web Resource.|accessdate=3 February 2023}} with respect to {{mvar|E}} are perspective. The polar triangle is bounded by the tangents to {{mvar|E}} at the vertices of {{math|△ABC}}. For example, the Trilinear polar of a point on the circumcircle must pass through its perspector, the Symmedian point X(6).

{{reflist}}

• Geometrikon page : [http://www.math.uoc.gr/~pamfilos/eGallery/problems/TrilinearPolar.html Trilinear polars]
• Geometrikon page : [http://www.math.uoc.gr/~pamfilos/eGallery/problems/IsotomicConicOfLine.html Isotomic conjugate of a line]

Category:Triangle geometry