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 {W

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:

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.

[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.

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 W

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.

[2] Honsberger, R.

[3] Lalesco, T.

The American Mathematical Monthly, Vol. 37, No. 3, (Mar., 1930), pp. 130-136

[4] Ramler, J. O.

Orthopole2.html

ParabolaChords.html

S_Triangles.html

Produced with EucliDraw© |