[alogo] 1. Orthopolar triangles

Two triangles ABC and RST inscribed in the same circle are called orthopolar when there is a point P which is the orthopole of any side of one triangle with respect to the other.
In S_Triangles.html we studied such triangles called S-Triangles. They are defined by considering a point P inside the deltoid-envelope of Wallace-Simson lines with respect to ABC. There are three tangents through P to this deltoid representing three Wallace lines {WR,WS,WT}. The respective points {R,S,T} are vertices of a triangle which is orthopolar to ABC.
The orthocenters of these two triangles are symmetric with respect to P. This is a characteristic property of orthopolar triangles. The two concepts are equivalent: S-Triangles are precisely orthopolar triangles. The discussion in S_Triangles.html proves that (S-Triangles => orthopolar). The other side of the equivalence is also obvious in view of the properties of orthopoles (especially these discussed in Orthopole2.html ).

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If ABC and RST are orthopolar, then their common orthopole P is intersection point of two tripples of Wallace lines. One tripple consisting of the Wallace lines of points {R,S,T} with respect to ABC and the other consisting of the Wallace lines of points {A,B,C} with respect to RST. By the discussion in Orthopole2.html and S_Triangles.html follows immediately that the two triangles are S-related.

[alogo] 2. The tangency to the parabola

Let the triangles ABC and RST inscribed in the same circle (c) be orthopolar.
[1] There is a unique point Q on the circle (c) for which the corresponding Wallace lines coincide.
[2] The two triangles have all their sides tangent to the parabola with directrix the line joining the orthocenters of the two triangles and focus the point Q.
[3] This is one more characteristic property of orthopolar triangles: They are inscribed in the same circle (c) and they are simultaneously tangent to a parabola.

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If the triangles ABC, RST are orthopolar then the line HH' joining their orthopoles defines a direction such that the corresponding Wallace lines with respect to these triangles which are parallel to HH' coincide. This follows trivially by finding the Wallace line W with respect to ABC which is parallel to HH'. Let Q be the corresponding point on the circumcircle (c) such that WQ=W. The Wallace line of the same point with respect to RST is parallel to W and passes through the middle of QH' hence coincides with W which does the same thing. By the well known properties of parabolas inscribed in triangles (see ParabolaChords.html ) the parabola with focus at Q and directrix the line HH' is tangent to the sides of both triangles.
Inversely, if the two triangles are inscribed in (c) and tangent to the same parabola as in the figura above, then their Wallace lines with respect to the focus Q of the parabola (which is necessarily on (c)) coincide and are parallel to line HH'. Thus there is a pair of parallel (coinciding) Wallace lines, which by the discussion in S_Triangles.html implies that the two triangles are S-related.

[alogo] 3. References

[1] Gallatly William The modern geometry of the triangle London, Francis Hodgson 1913, p.49.
[2] Honsberger, R. Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington DC, Math. Assoc. Ammer., 1995, pp. 106-110.
[3] Lalesco, T. La Geometrie du Triangle. Paris, Jacques Gabay, 1987, p. 17.
The American Mathematical Monthly, Vol. 37, No. 3, (Mar., 1930), pp. 130-136
[4] Ramler, J. O. On Triangles Having a Common Mean The American Mathematical Monthly, Vol. 47, No. 3, (Mar., 1940), pp. 140-145

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