[2] The Simson line of P with respect to the incircle is orthogonal to the direction of X

[3] Point P is the isogonal/isotomic conjugate of the point at infinity X

[4] The isogonal/isotomic conjugate of X

[1] Start with point P and the tangent to the incircle of ABC at this point. Quadrilateral PC''A''B'' is inscribed in the circle hence by Pascal's theorem (see PascalOnQuadrangles.html ) the intersection points of opposite sides {B

[2] Because of the parallels {A''B'',A''C''} to the sides of ABC each of the triangles {AB''C

[3] Obviously CC

[4] Consider quadrilateral BCB

[5] The bundle of lines B(C,C

(i) S is the tripole of line B

(ii) Lines {B

(iii) Line AA

[6] Since (B

In the file HyperbolaPropertyParallels.html the same result is obtained in the reverse way, by starting from the point at infinity and landing down to P.

The statement on the Simson line is immediate and follows by an angle chasing argument indicated in the figure below. The figure is a magnification of the previous one. The small circle passes through points P and B'' and the two projections of P on the sides of A''B''C''. It results by the definition of the Simson line of P with respect to A''B''C''. The angles indicated at P are equal because angle(A''PB'') is 60 degrees as is also the angle of the cyclic quadrilateral at P opposite to B''.

The fact that P is the isogonal conjugate of the point at infinity X

The statement on the anti-homothety F interchanging ABC and A''B''C'' follows from the previous result and the fact that t(P') = t(F(P)) = F(t(P)) = F(X

[2] The perspector is the point at infinity X

[3] The tripole of line B

[4] Segments {A

[5] Define the affinity H of the plane by the properties :

(i) H fixes the points of line PA*.

(ii) Every point Q is mapped to Q'=H(Q) such that QQ' is parallel to X

The affinity H can be interpreted as a perspectivity with center at the point at infinity defined through X

[1,2] are consequences of the discussion in section-1. Because of the harmonicity of tetrads (S,C

Brianchon2.html

CircumconicsTangents.html

Desargues.html

HyperbolaPropertyParallels.html

HyperbolaPropertyParallels2.html

IncircleTangents2.html

PascalOnQuadrangles.html

IsogonalOfCircumcircle.html

IsogonalGeneralized.html

TriangleConics.html

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