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K742

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X(1), X(5), X(155), X(185)

excenters

vertices of the orthic triangle

Ea = HbHc /\ AX(155), Eb and Ec likewise

Let M be a fixed point. The locus of point P such that the line MP is perpendicular to the trilinear polar of the isogonal conjugate of P is a rectangular hyperbola H(M) passing through X6, X2574, X2575, M hence homothetic to the Jerabek hyperbola.

H(M) is a bicevian conic if and only if M lies on K742 in which case the perspectors P1, P2 lie on the circum-conic with center X(6) and the isogonal transform of the Stammler hyperbola respectively.

For example, with M = X(1), X(5), X(185) we find the bicevian rectangular hyperbolas B(X651, X1), B(X110, X2), B(X648, X4) respectively.

K742 is a psK with respect to the orthic triangle since the tangents at Ha, Hb, Hc concur at X(235) but not lying on the cubic.

When isogonal is remplaced with isotomic, we obtain the analogous cubic K461.