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A', B', C' : antipodes of A, B, C on the circumcircle

Q1, Q2, Q3 : vertices of the Thomson triangle

infinite points of its sidelines

other points below

Consider the three parabolas Pa, Pb, Pc with foci A, B, C and directrices BC, CA, AB respectively. These parabolas generate a net N of conics and any conic of the net may be represented under the form p Pa + q Pb + r Pc where p, q, r are not all zero. Further details below.

K703 is the Jacobian of the net i.e. the locus of M such that the polar lines of M in Pa, Pb, Pc (or any other system of three independent conics of the net) concur at M' which is also a point of K703.

The reflection of K703 about O is an isogonal nK with root X(3618). It follows that K703 meets the sidelines of A'B'C' at three collinear points.

K703 is also a nK with respect to the Thomson triangle with root its centroid X(3524).


Prehessians and consequences

K703 has three (always real) prehessians Pr_1, Pr_2, Pr_3.

It follows that one can find three involutions F_1, F_2, F_3 on K703 : each point M on K703 is mapped to the center F_i(M) of its (degenerate) polar conic in Pr_i.

Each involution F_i has three singular points which are the vertices of a triangle T_i inscribed in K703.

One of these, say T_1, is the Thomson triangle. The two other and the triangle bounded by the asymptotes are homothetic to T_1.


Points on the altitudes and consequences

The polar lines of A' in Pb, Pc coincide in the altitude AH and it is easy to identify its three common points with K703.

Two are the intersections Ha1, Ha2 with the circle C(A, 2R) and the third Ha3 is the reflection of the foot of AH in the midpoint of AH.

With the three involutions above, we find the following points on K703 :

• Ha12 = F_2(Ha1) = Ha21 = F_1(Ha2),

• Ha11 = F_1(Ha1) = Ha22 = F_2(Ha2),

• Ha31 = F_1(Ha3) and Ha33 = F_3(Ha3),

Note that :

• Ha32 = F_2(Ha3) = A',

• Ha23 = F_3(Ha2) = Ha1,

• Ha13 = F_3(Ha1) = Ha2.

The two other altitudes yield two sets of analogous points.

The polar conic PC(Ha33) of Ha33 in K703 contains Ha33 and Ha1, Ha2, Ha11, Ha12 hence the tangents at these four latter points concur at Ha33. The diagonal triangle of the quadrilateral is A', Ha3, Ha31 and then K703 is a pivotal cubic with pivot Ha33 in this triangle. It follows that the tangents at A', Ha3, Ha31 must concur at the isopivot Ha0 whose polar conic PC(Ha0) also contains Ha33.


The net N of conics generated by the three parabolas

N contains :

• a circle with center X(20) and radius √3 times that of the polar circle, real when ABC is obtusangle,

• a pencil of rectangular hyperbolas with centers on C(X3, 3R),

• a family of parabolas when p:q:r lies on the Steiner ellipse,

• a family of decomposed conics when p:q:r lies on a cubic passing through the vertices of the cevian triangles of X(75), X(76) and through the traces of the trilinear polar of X(75).