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X(3), X(4), X(155), X(1498), X(6759), X(7387), X(9833), X(68389)

infinite points of the altitudes

Ka, Kb, Kc : vertices of the tangential triangle

La, Lb, Lc : their reflections in X(6759)

A', B', C' : vertices of the Aries triangle

Ha, Hb, Hc : their reflections in X(6759), 3rd points on the altitudes

Geometric properties :

(a communication by César Lozada, 2025-04-26)

K1398 is the Spieker central cubic K033 of the tangential triangle. It is a central pivotal cubic with center X(6759) in this triangle. Its pivot is X(1498) and its isopivot is X(4).

Locus property

Let P be a point with pedal triangle PaPbPc. The circle with center Pa, passing through A, meets BC at A1, A2. Define B1, B2 and C1, C2 cyclically. These six points are conconic if and only if P lies on K1398.

When P = X(155), the conic is a circle, namely the Dou circle.

X(155) is the orthocenter of the tangential triangle and also the center of this circle.

In this case, the angles A1AA2, B1BB2, C1CC2 are all right angles.

The Dou circle is described in Jordi Dou, Problem 1140, Crux Mathematicorum, 28 (2002), 461-462.