Harmonic_Perspective

1. Inscribed in triangle perspective triangles

The basic configuration here is:
(1) a triangle ABC,
(2) a line (e) intersecting the sides of the triangle at points A*(BC), B*(CA) and C*(AB).
(3) Finally there is a point O not lying on the sides of ABC and not lying also on e. Join O to points A*,B*,C* and define correspondingly the intersections with the other sides of the triangle: A'=(BC,OB*), A''=(BA,OB*), B'=(CA,OC*), B''=(CB,OC*), C'=(AB,OA*), C''=(AC,OA*).

2. The perspectivity is harmonic

The two triangles A'B'C' and A''B''C'' are, by their proper definition, perspective with respect to the point O. Hence by Desargues theorem (see Desargues.html ), their corresponding sides intersect on the perspectivity axis d containing the intersection points A₀=(B'C',B''C''), B₀=(C'A',C''A'') and C₀=(A'B',A''B''). The two triangles are also perspective in the sense of projectivities, i.e. there is a perspectivity f with the same center and axis as those of the Desargues theorem mapping A'B'C' to A''B''C''.
The perspectivity is harmonic. This means that corresponding points, as for example (A',A'') are harmonic conjugate with respect to (O,T), where T is the intersection point of line A'A'' with the perspectivity axis.
This claim follows from the characterization of such perspectivities by means of the existence of a conic c passing through all six of the vertices of triangles A'B'C' and A''B''C'' ([Lachlan, p. 118]) (see PerspectivityAndPerspectiveTriangles.html ). To show that such a conic exists apply the inverse of Pascal's theorem to the hexagon:
C'A''A'B''B'C''.
By assumption, the pairs of opposite sides of this hexagon:
C*= (C'A'', B''B'), A*= (C''C',B''A') and B* = (B'C'', A''A'),
define collinear points. Hence the hexagon can be inscribed in a conic c as claimed.

3. The associated conics

By the general properties of harmonic perspectivities (see PerspectivityAndPerspectiveTriangles.html ), the vertices of the two triangles A'B'C' and A''B''C'' lie on a conic c. The perspectivity center O and perspectivity axis d of the two triangles are corresponding pol and polar with respect to this conic.
The two triangles A'B'C' and A''B''C'' having their vertices on the same conic c implies that they have also their sides tangent to another conic c'. This follows from Poncelet's theorem (see Poncelet.html ), by constructing the conic c' tangent to the three sides of A'B'C' and the two sides of A''B''C''. By the aforementioned theorem c' will be tangent also to the third side of A''B''C''.
The intersection points of the sides of the two triangles A'B'C' and A''B''C'' define also the vertices of an hexagon inscribed in a conic. This because the pairs of opposite sides intersect at the points A₀, B₀, C₀ of line d. The result follows by applying Pascal's theorem.

4. Special cases

In general the associated conics described above do not belong to the same family of conics. In special cases though it may happen that all these conics belong to the same family of conics. This is the case, for example, when line e and point O are correspondingly tripolar and trilinear pol with respect to the triangle ABC. The corresponding configuration is to be seen in the file TrilinearFamilyConics.html .

5. Line and point from the conic

The argument proving the existence of the conic c in nr. 2 is reversible and shows that every conic intersecting the sides of a triangle at points A',B',C',A'',B'',C'', as in the figure, defines lines A'A'', B'B'' and C'C'' intersecting the corresponding sides of ABC at points B*,C*,A* lying on a line e. In general the two triangles A'B'C' and A''B''C'' are not (harmonically) perspective. This happens only for particular conics with respect to the triangle ABC.

6. Twin triangles

Fixing the triangle of reference ABC, the construction described in nr. 1 defines two harmonically perspective triangles, which I often call twins generated by the point O and the line e.

Bibliography

[Lachlan] Lachlan R. An elementary treatise on modern pure Geometry. London, Macmillan, 1893

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