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Trilinear polarity

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Axis of perspectivity of a given triangle, its cevian triangle, and some point

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." 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.

Definitions

Construction of a trilinear polar of a point P   Given triangle △ABC   Cevian triangle △DEF of △ABC from P   Cevian lines which intersect at P   Constructed trilinear polar (line XYZ)

Let △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.

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.

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 equation

Let the trilinear coordinates of the point P be p : q : r. Then the trilinear equation of the trilinear polar of P is

x p + y q + z r = 0. {\displaystyle {\frac {x}{p}}+{\frac {y}{q}}+{\frac {z}{r}}=0.}

Construction of the trilinear pole

Construction of a trilinear pole of a line XYZ   Given trilinear polar (line XYZ)   Given triangle △ABC   Cevian triangle △UVW of △ABC from XYZ   Cevian lines, which intersect at the trilinear pole P

Let 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 polars

Some of the trilinear polars are well known.

Poles of pencils of lines

Animation illustrating the fact that the locus of the trilinear poles of a pencil of lines passing through a fixed point K is a circumconic of the reference triangle.

Let 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

x X + y Y + z Z = 0. {\displaystyle {\frac {x}{X}}+{\frac {y}{Y}}+{\frac {z}{Z}}=0.}

Since this passes through K,

x 0 X + y 0 Y + z 0 Z = 0. {\displaystyle {\frac {x_{0}}{X}}+{\frac {y_{0}}{Y}}+{\frac {z_{0}}{Z}}=0.}

Thus the locus of P is

x 0 x + y 0 y + z 0 z = 0. {\displaystyle {\frac {x_{0}}{x}}+{\frac {y_{0}}{y}}+{\frac {z_{0}}{z}}=0.}

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 K is a circumconic E of the triangle of reference.

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

References

  1. ^ Coxeter, H.S.M. (1993). The Real Projective Plane. Springer. pp. 102–103. ISBN 9780387978895.
  2. Coxeter, H.S.M. (2003). Projective Geometry. Springer. pp. 29. ISBN 9780387406237.
  3. Weisstein, Eric W. "Trilinear Polar". MathWorld—A Wolfram Web Resource. Retrieved 31 July 2012.
  4. Weisstein, Eric W. "Trilinear Pole". MathWorld—A Wolfram Web Resource. Retrieved 8 August 2012.
  5. Weisstein, Eric W. "Perspector". MathWorld—A Wolfram Web Resource. Retrieved 3 February 2023.
  6. Weisstein, Eric W. "Polar Triangle". MathWorld—A Wolfram Web Resource. Retrieved 3 February 2023.

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