IncircleTangents

1. Incircle tangents or the beauty of the equilateral

[1] The tangent to the incircle of an equilateral triangle at a point P is the trilinear polar of a point at infinity X_P, determined by a direction of paralles X.
[2] The Simson line of P with respect to the incircle is orthogonal to the direction of X_P.
[3] Point P is the isogonal/isotomic conjugate of the point at infinity X_P with respect to the medial triangle A''B''C''.
[4] The isogonal/isotomic conjugate of X_P with respect to triangle ABC is point P' anti-homothetic of P in the anti-homothety which interchanges ABC with A''B''C''.

[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 {B₃,C₃} and of tangents at opposite vertices {A,A*} are on a line and make a harmonic division.
[2] Because of the parallels {A''B'',A''C''} to the sides of ABC each of the triangles {AB''C₃, B₃C''A} is similar to B₃A''C₃, hence they are similar to each other. It follows that B''C₃/AB''=AC''/B₃C'' and since {B'',C''} are the middles of sides we get B''C₃/B''C=BC''/C''B₃, implying that triangles {B₃C''B, CB''C₃} are similar. Thus lines {CC₂,BB₂} are parallel.
[3] Obviously CC₂=2CC₃ and BB₂=2BB₃. Hence line B₂C₂ passes through A*.
[4] Consider quadrilateral BCB₁C₁ which circumscribs the incircle and take polars with respect to the incircle. By Brianchon's theorem for quadrangles (see Brianchon2.html ) its diagonals {BB₁,CC₁} intersect at the same point S at which intersect lines {PA'',C''B''} joining contact points of opposite sides. Since C''P is the polar of C₁ and passes through C₃, the polar of C₃ will pass also through C₁. Analogously it will pass through C. Hence C₁C coinsides with the polar of C₃ and will pass through B₃. Analogously BB₁ passes through C₃. Thus B₃C₃ is the polar of S.
[5] The bundle of lines B(C,C₃,C₂,B₃) is harmonic (since CC₃=C₃C₂). Hence defines on AC a harmonic division. This implies that B₁ is the harmonic conjugate of B₂ with respect to {A,C}. Analogously C₁ is the harmonic conjugate of C₂ with respect to {A,B}. This implies:
(i) S is the tripole of line B₂C₂.
(ii) Lines {B₁C₂,C₁B₂} intersect at a point A₂ on line BC.
(iii) Line AA₃ is parallel to lines BB₃ and CC₃.
[6] Since (B₁,B₂,A,C) and (C₁,C₂,A,B) are harmonic divisions line B₁C₁ is the tripolar of the point at infinity determined by the parallel lines {BB₂,CC₂} (apply the definition of tripolar).

Remark The reasoning here is to work from point P on the circumcircle and the tangent at P up to the tripole of this line with respect to ABC, which is the point at infinity determined by the direction BB₂.
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_P with respect to triangle A''B''C'' follows from the discussion on isogonal conjugates of points of the circumcircle in IsogonalOfCircumcircle.html .
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_P) = X_P. Last equation being the obvious invariance of the direction of parallels by the anti-homothety F. The fact that t*F = F*t (* denoting the composition of maps) is discussed in some extension in IsogonalGeneralized.html .

2. Two perspective triangles

[1] Triangle SB₃C₃ is self-polar with respect to the incircle of ABC, perspective to it and its side-lines pass through the vertices of ABC.
[2] The perspector is the point at infinity X_P and the line of perspectivity ( Desargues.html ) is the tangent PA* at P.
[3] The tripole of line B₁C₁ with respect to triangle SB₃C₃ is the same point at infinity determined by X_P.
[4] Segments {A₀A₂,B₀B₂,C₀C₂} are correspondingly the double of {AA₀,BB₀,CC₀}. Points {A₀,B₀,C₀} denote correspondingly the intersection points of the parallel lines {AA₂,BB₂,CC₂} with the tangent PA*.
[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_P and line PA* divides QQ' in ratio 2, i.e. the intersection point Q₀ of QQ' with PA* satisfies QQ₀/Q₀Q'=2.
The affinity H can be interpreted as a perspectivity with center at the point at infinity defined through X_P, axis the line PA* and homology coefficient k=-0.5. It maps triangle ABC to triangle SB₃C₃.

[1,2] are consequences of the discussion in section-1. Because of the harmonicity of tetrads (S,C₃,B,B₁) etc. the tripole of line B₁C₁ with respect to triangle SB₃C₃ is also the point at infinity determined by X_P. This proves [3]. [4] is a consequence of [3] of section-1. [5] is a consequence of [4].

Remark These facts generalize to a property for self-polar triangles with respect to a conic. See the relative discussion in Autopolar2.html .

1. Incircle tangents or the beauty of the equilateral

2. Two perspective triangles

See Also