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vertices of the circumtangential triangle.

other points described below.

see also other figures at the bottom of the page.

For any point M on Kjp = K024 = nK0+(X6, X6), the circumcevian triangle of M and ABC are parallelogic. See K024, property 8. One of the centers of parallelogy is M* (the isogonal conjugate of M) and the other M' is a point on K409 (Jean-Pierre Ehrmann, private message).

M' is also the perspector of the circumcevian triangles of M and M*. See K024, property 9.

K409 is the isogonal transform of K198, an isotomic nK60. As such, K409 is a nK with pole X(32), root X(2). Hence it has three real asymptotes parallel to the sidelines of ABC. These asymptotes form a triangle homothetic of ABC under the homothety with center X(32), ratio 1 / (1-cos2w) where w is the Brocard angle. K409 meets its asymptotes on a parallel to the trilinear polar of X(32) which is the satellite line of the line at infinity.

K409 is also a CircumTangential cubic. See Table 25. Thus, it meets the circumcircle at the same points as K024 and the three remaining common points of the two cubics lie on the trilinear polar of X(83), the perpendicular at X(23) to the Euler line.


K409 meets the sidelines of the antimedial triangle at A, B, C, the infinite points of the sidelines of ABC and three other points on the circum-conic with perspector X(1506), the intersection of the lines X(2)X(32) and X(5)X(39).

We know that K024 meets the Lemoine axis at the centers of the Apollonian circles. K409 meets this same line at the same points as the isogonal transform of the cubic K093.

This transform is nK0(X32, X9463) where X9463 is the intersection of the lines X(2)X(6) and X(32)X(110).