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∑ b^2 c^2 x^3 (y+z) [c^2 y (x-y+z) - b^2 z(x+y-z)] = 0

X(1), X(3), X(4)


midpoints of ABC

anti-points : imaginary points on the perpendicular bisectors, also on K003 and K024, see below

infinite points of the 6 bisectors

vertices of the two circum-perp triangles

centers of the Apollonius circles

points of K020 on the Lemoine axis

their isogonal conjugates on the Steiner ellipse (S)

Let P be a point in the plane of a triangle ABC, such that P is not situated on any of its sidelines, nor on its bisectors, nor on the circumcircle, nor on the line of infinity.

Let La, Lab, Lac be the radical axes of the circle with center P passing through A, relatively to the circles (PBC), (PCA), (PAB).

Let Ab = La ∩ Lab, Ac = La ∩ Lac and A' = BAb ∩ CAc. Define B' and C' cyclically.

The lines AA', BB', CC' are concurrent (actually parallel) if and only if P is on the cubic K024 (Angel Montesdeoca, 2021-09-08).

More generally, these lines AA', BB', CC' bound a triangle perspective (at Q) to ABC if and only if P lies on K024 (as above with Q at infinity) or P lies on Q169.


Properties of Q169

A, B, C are nodes with tangents the symmedians and the sidelines of the antimedial triangle.

The tangents at the in/excenters also pass through X(6).

The tangents at O and H pass through X(66).

Q169 has six real asymptotes parallel to the bisectors at the midpoint of the corresponding median.

The isogonal conjugates of their infinite points are the vertices of the two circum-perp triangles A1B1C1 and A2B2C2, obviously on the circumcircle of ABC and on these bisectors.

Q169 meets the Lemoine axis (trilinear polar of the Lemoine point) at the centers Ωa, Ωb, Ωc of the Apollonius circles and three other points which lie on K020.

The isogonal conjugates of these latter three points lie on Q169, the Steiner ellipse and K020.

Q169 meets K024 at A, B, C (each counting for three), Ωa, Ωb, Ωc and three pairs of Cayley's anti-points, on the perpendicular bisectors, on K003, These points are imaginary, two by two isogonal conjugates and symmetric about a midpoint of ABC, two by two collinear with one center Ωa, Ωb, Ωc. They are the common points (apart A, B, C) of the cevian lines of the circular points at infinity.