Permuted coordinates of points

Parallels are affine invariant. So are Cevian ratios. Everything on this page will be as well.

This is a topic on which we are blessed with a good set of resources. Bernard Gibert has a wonderful article on this subject. Kapetis' first volume (in Greek) also treats this topic well. Paris Pamfilos wrote an equally wonderful article for FG, but he did not call it rotated or permuted points, rather the "action of D3 on the triangle."

Choose a point (l:m:0) on the side c. Let it divide the side in ratio v, taken counterclockwise. The parallels drawn as shown close to form a hexagon. Since parallels preserve ratios each vertex has the ratio v or 1/v, its isotomic. Hence lines from vertex C to the two points on edge c are isotomic.

The vertices of the hexagon are on an ellipse centered at G and of the same shape and inclination as the Steiner ellipse.

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We can follow this logic with a point not on an edge. Like Kapetis we indicate a point by its Cevian ratios. Again following Kapetis (I think) I call these logocentric coordinates <l , m , n> with lmn = 1. These coordinates are not homogeneous. We will use barycentric coordinates as well.

A point P has three traces on the edges. For each of the three traces, follow parallels around the sides in the counterclockwise direction as shown in the figure. Now connect these three traces to construct the point P_< = <m, n, l>. Go in the clockwise direction to construct P_> = < n, l , m >. We call P_< and P_> the rotated versions of P.

Similarly there are points formed from permutations of the coordinate rather than rotations of them, which we denote as P₂₃, P₃₁, P₁₂. Their construction will be seen in a paragraph or so.

From these six points, there are two natrual triangle P P_< P_> , and P₂₃ P₃₁ P₁₂.

Each of these is triply in perspective at tP₂₃ tP₃₁ tP₁₂ and tP tP_< tP_> where t is the isotomic conjugate operation.

All of these triangles have centroid G.

Table of barycentric coordinates

P	P>	P>	P23	P31	P12
(l:m:n)	(n:l:m)	(m:n:l)	(l:n:m)	(n:m:l)	(m:l:n)
tP	tP>	tP>	tP23	tP31	tP12
(mn:nl:lm)	(lm:mn:nl)

Notice these parallelisms (dottet lines) among the lines like P—P_> || P_<—P₁₂ || CA.

Consider this harmonic system of points

P_< P_< + P₁₂ P₁₂ P_< – P₁₂ ~ ( 0 : 1 : –1 )

Since the last of these is at infinty, must be the midpoint, P_< + P₁₂ which we conclude is on a median.

P_< P_< + P_> P_> P_< – P_> ~ ( m–n : n–l : l–m )

Since the last of these is the infinite point on the dual of P, the second must be the midpoint, P_< + P_> ~ mP and the line P_<—P_> is parallel to the dual of P.

Rotated Tripolars

Each of the permuted points has a tripolar. The tripolars of tP> and tP< concur at P2-, the inverse of P in the Steiner ellipse.