Home page | Catalogue | Classes | Tables | Glossary | Notations | Links | Bibliography | Thanks | Downloads | Related Curves


too complicated to be written here. Click on the link to download a text file.

X(3), X(194)

infinite points of K003

P1, P2, P3 : points of K003 on the Lemoine axis

T1, T2, T3 : vertices of the CircumTangential triangle on K024

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

imaginary foci of the Brocard ellipse i.e. common points of the Brocard axis and the Kiepert hyperbola

other points and details below

Geometric properties :

A'B'C' is the only triply bilogic triangle inscribed in the circumcircle, in short the ITB triangle. See K003 and the Steiner ellipse.

K1098 is the McCay cubic in this ITB triangle whose centroid is X(7709), the radial center the stelloid. See K1101, another related stelloid.

Recall that X(3), X(194) are the circumcenter and orthocenter. Note that ABC and A'B'C' also share the same Lemoine point X(6) hence the same Brocard axis. The Brocard ellipse is inscribed in both triangles.

Some reminders and further comments :

• A', B', C' are the common points of (O) and K410, K411, the rectangular hyperbola passing through X(32), X(194), X(511), X(512), X(805), etc. The tangents to K1098 at these points and at X(3) concur at X(194) since it is the isopivot of the cubic.

• A'B'C' is triply perspective to ABC at P1, P2, P3. The tangents to K1098 at these points concur at X(2) hence X(2) must lie on K1099, the Hessian cubic of K1098. The polar conic of X(2) in K1098 splits into the Lemoine axis (L) and the parallel at X(376) to the Brocard axis.

• ABC is triply orthologic to A'B'C' at O1, O2, O3 which are the isogonal conjugates of P1, P2, P3 with respect to ABC. These points lie on the Steiner ellipse (S) and on K003.

• A'B'C' is triply orthologic to ABC at Q1, Q2, Q3 which are the isogonal conjugates of P1, P2, P3 with respect to A'B'C'. These points lie on the Steiner ellipse (S') of A'B'C' and on K1098 since it is K003 for A'B'C'. Note that the axes of (S) and (S') are parallel at X(2) and X(7709) respectively to the asymptotes of the Kiepert hyperbola. Also (S') is homothetic to the circum-conic passing through X(99), X(685). The union of the axes of (S') is actually the polar conic of X(511) in K1098.

• It follows that X(3), Pi, Oi, Qi (i = 1, 2, 3) are collinear.


K1098 obviously passes through the vertices U, V, W of the cevian triangle of O with respect to A'B'C' and the in/excenters of A'B'C'. These four points Jo (incenter) and Ja, Jb, Jc (excenters) lie on K1100, K1102 and (H), the rectangular hyperbola with center X(98) passing through X(4), X(40), X(376), X(1670), X(1671), X(3413), X(3414), X(7709), X(12251).

The two remaining common points of (H) and K1098 are complex conjugates on the line parallel at X(194) to the Brocard axis which is the polar line of O in (H).

(H) is the Wallace hyperbola of A'B'C' and in fact the reflection in O of the Wallace hyperbola (W) of ABC. Note that (W) is the anticomplement of the Kiepert hyperbola, the polar conic of X(194) in K410 and more generally the polar conic of P in pK(X6, P) for every P on (W) e. g. K002, K004.