In the following properties tripoles/tripolars are taken with respect to triangle ABC. Both operations are denoted by tr. Thus, tr(P) the tripolar of a poing is a line and tr(L) the tripole of a line is a point.

[1] Lines B'C', for various directions of the parallels, envelope an hyperbola.

[2] The hyperbola has b, c as asymptotic lines. The varying triangle AB'C' is an asymptotic one of the hyperbola and, as such, has area constant and equal to that of triangle ABC.

[3] Let B*(resp. C*) be the harmonic conjugate of B' (resp. C') with respect to AB (resp. AC). Then line B*C* is the trilinear polar, with respect to triangle ABC, of the point at infinity determined by the direction of the parallel lines.

[4] Lines B'C' and B*C* intersect on line BC at a point A*. Also the pair of lines C'B* and B'C* intersect on line BC at the harmonic conjugate A

[5] Let Q be the intersection point of line BC' with B*C*. The length-ratio C'Q/QB = 2.

[6] The tripoles S=tr(L

[7] Line A

[8] Lines BC' and CB', joining the non-collinear with the center (of hyperbola) vertices of

The proofs are very simple. The area of the varying triangles AB'C' is equal to the area of the fixed triangle ABC (see Trapezium.html ). [1,2] follow by applying the remark made in AsymptoticTriangleInv.html .

Regarding the proof of [3], note that line B*C* satisfies the definition of the trilinear polar for the point at infinity (denoted in the figure by [P]) defined by the direction of parallel lines (which is the direction of BP).

Regarding [4], the statement on the coincidence of A* on BC follows from the fact that (B,A,B*B') and (C,A,C*,C') are two harmonic divisions on lines b, c and have A in common (see CrossRatioLines.html ). The same argument proves the coincidence of B'C* and C'B* at A

[5] follows from the previous arguments and the fact that (B,M,Q,C') is a harmonic division, where M denotes the middle of BC' (the bundle of lines A*(B,M,Q,C') is a harmonic one).

To see [6] note first that the middles {M,M'} of {BC',CB'} are collinear with {A,A*} and build with them a harmonic division (consider the complete quadrilateral-trapezium BCB'C'). Then, {M,B*,C} are collinear as well as {B,C*,M'}. Also by definition the tripole S of B'C' is the intersection of the diagonals {MC,M'B} of trapezium BCM'M. Thus S is on line AA

To prove [7] note that line bundles S(B,C,A

In the file HyperbolaPropertyParallels2.html I discuss the figure resulting from the previous one by applying to it a projectivity fixing the vertices of the triangle and mapping the centroid G to an other arbitrary point of the plane.

[2] Of particular importance is the quadrangle A

the tripolars tr([P]) of points at infinity [P] are tangents to the Steiner inconic.

[4] Of some importance is also the collinearity {A

(i) Join A to [P] with a line and find its intersection S with B

(ii) join A

[5] The above discussion is about the relations between the inner Steiner ellipse, which is the inconic of ABC with respect to its centroid G and the (three) inconics having perspectors the

[6] One could discuss the subject in an inverse way. Starting with a point A

CrossRatioLines.html

HyperbolaPropertyParallels2.html

IncircleTangents.html

PascalOnQuadrangles.html

Trapezium.html

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