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X(1), X(20), X(64), X(1350), X(3424), X(4334), X(5373), X(6194), X(7220)

excenters of ABC

A', B', C' : excenters of the Thomson triangle, the incenter being X(5373)

vertices of the cevian triangle of X(1350)

imaginary foci of the Brocard inellipse.

(contributed by Peter Moses, 2016-03-12)

K763 is a member of the pencil of isogonal pKs whose pivot lies on the line X(20), X(64), X(69), X(253), X(489), X(490), X(1043), X(1350), X(1503), X(1975), X(2897), etc. All these cubics pass through A, B, C, X(20), X(64) and the in/excenters of ABC.

K763 = pK(X6, X1350) is the one which also passes through the in/excenters of the Thomson triangle.

This pencil contains K004 = pK(X6, X20), K169 = pK(X6, X69), K270 = pK(X6, X1503) which is circular, K364 = pK(X6, X14516).

pK(X6, X253) is another notable member passing through X(1), X(20), X(64), X(154), X(253), X(1073), X(1249).


K763 also belongs to the pencil of isogonal pKs with pivot on the Brocard axis containing K003 and K102.


K615 and K763 generate a pencil of circum-cubics passing through X(64), X(3424) and the four in/excenters of the Thomson triangle.

Each cubic K(P) can be associated to a point P on the line GK = X(2)X(6) such that GP = t GK where t is a real number or infinity. In this case, its equation takes the form K(P) = (1 - t)(a^2 + b^2 + c^2) K615 + 3t K763.

Obviously, K(X2) = K615 and K(X6) = K763.

This pencil contains one circular cubic obtained for t = ∞ i.e. P = X(524).


K(P) meets the line at infinity at the same points as pK(X6, P∞) where P∞ is the homothetic of X(1350) under h(X20, t). The six remaining common points lie on the Jerabek hyperbola namely A, B, C, X(64) and two other points on a line passing through G. The third point of K(P) on this latter line also lies on the rectangular hyperbola passing through G and the in/excenters of the Thomson triangle. This hyperbola is homothetic to the Kiepert hyperbola.

K(P) meets the circumcircle (O) at the same points as pK(X6, Po) where Po is the homothetic of X(1350) under h(X2, t). The three remaining common points lie on the Euler line.

Hence, for a given point P on GK, Po is the intersection of the line X(2)X(1350) and the parallel at P to the Brocard axis. P∞ is the intersection of the line X(20)X(1350) and the parallel at Po to the Euler line.

When P traverses the line GK, the line PP∞ envelopes the parabola (P) with focus X(691), directrix the line (D) = X(99)X(110).

(P) contains X(182), X(376), X(542), X(1992) and is tangent at these points to the Brocard axis, the Euler line, the line at infinity, the line GK respectively.


The figure opposite shows K615, K763 together with the (green) circular cubic K(X524) and the rectangular hyperbola (H) which is the reflection about O of the Stammler hyperbola.

(H) share five common points with each cubic K(P) of the pencil namely X(64), X(5373), A', B', C'. Hence it must have another (always real) common point which is X3 for K615, X1350 for K763.

The one on the circular cubic is X(16010), the reflection of X(2930) about X(3).