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(1), X(6), X(13), X(14), X(399), X(2574), X(2575) excenters foci of the Kellipse (inellipse with center K when the triangle ABC is acute angle) points of pK(X6, X323) on (O) 

Let L be a line passing through X(3) and M a variable point on L. Denote by J(M) the rectangular hyperbola passing through M, X(3), X(6), X(2574), X(2575), hence homothetic to the Jerabek hyperbola. J(M) is a member of the Jerabek pencil, see below. Denote by K(M) the isogonal pivotal cubic with pivot M i.e. pK(X6, M). J(M) and K(M) meet at M and five other points which generally lie on a circular quartic Q(L), together with the line L. Trivial cases : • when L is parallel to one asymptote of the Jerabek hyperbola, J(M) splits into L and its perpendicular at X(6). • when L is the Brocard axis, J(M) splits into this axis and the line at infinity. In both cases, J(M) is independent of M and Q(L) is not a quartic but J(M) itself. This is excluded in the sequel. Properties : When L rotates about X(3), Q(L) contains 16 fixed points therefore these quartics are in a same pencil. These points are : • X(2574), X(2575), two circular points at infinity, • A, B, C, the four in/excenters, X(6), the four foci of the inconic with center X(6) sometimes called ellipseK when ABC is acutangle. Furthermore, Q(L) meets the circumcircle (O) again at the same (not always real) points as pK(X6, Z) where Z is a point on the line X(2)X(6), the barycentric product of X(99) and the infinite point of a perpendicular to L. Examples : • when L is the Euler line, Q(L) is Q002, Z = X(2), pK(X6, Z) is K002 and Q(L) meets (O) at the vertices of the Thomson triangle. • when L is the line through X(110), Q(L) is Q112, Z = X(323). 

Recall that the members of the Jerabek pencil are the rectangular hyperbolas passing through X(3), X(6), X(2574), X(2575). Their centers Ω lie on the line containing X(2), X(98), X(110), X(114), X(125), X(147), X(182), X(184), X(287), X(542), X(1352), X(1899), X(1976), X(2001), X(3047), etc. • Ω = X(542), union of the line at infinity and the Brocard axis. • Ω = X(125), the Jerabek hyperbola itself, the only circumconic. • Ω = X(110), the Stammler hyperbola, the only diagonal conic. • Ω = X(5642), the Jerabek hyperbola of the Thomson triangle. • Ω = X(5972), the complement of the Jerabek hyperbola, the bicevian conic C(X2, X110). These hyperbolas are the polar conics of X(3) in a cubic of the Euler pencil or the polar conic in K003 of a point on the Euler line. Further details in the page Q002.

