| 词条 | Trilinear polarity |
| 释义 |
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."[1] It was Poncelet (1788–1867), a French engineer and mathematician, who introduced the idea of the trilinear polar of a point in 1865.[1][2] DefinitionsLet 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.[1] 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.[1] 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. Trilinear equationLet the trilinear coordinates of the point P be (p : q : r). Then the trilinear equation of the trilinear polar of P is[3] x / p + y / q + z / r = 0. Construction of the trilinear poleLet 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 trilinear polarsSome of the trilinear polars are well known.[4]
Poles of pencils of linesLet P with trilinear coordinates ( X : Y : Z ) be the pole of a line passing through a fixed point K with trilinear coordinates ( x0 : y0 : z0 ). Equation of the line is
Since this passes through K,
Thus the locus of P is
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. References1. ^1 2 3 {{cite book|last=Coxeter|first=H.S.M.|title=The Real Projective Plane|year=1993|publisher=Springer|isbn=9780387978895|pages=102–103}} 2. ^{{cite book|last=Coxeter|first=H.S.M.|title=Projective Geometry|year=2003|publisher=Springer|isbn=9780387406237|pages=29}} 3. ^{{cite web|last=Weisstein|first=Eric W.|title=Trilinear Polar|url=http://mathworld.wolfram.com/TrilinearPolar.html|publisher=MathWorld—A Wolfram Web Resource|accessdate=31 July 2012}} 4. ^{{cite web|last=Weisstein|first=Eric W.|title=Trilinear Pole|url=http://mathworld.wolfram.com/TrilinearPole.html|publisher=MathWorld—A Wolfram Web Resource.|accessdate=8 August 2012}} External links
1 : Triangle geometry |
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