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X(2), X(4), X(69), X(263), X(13428), X(13439), X(16770), X(16771), X(17751), X(19778), X(19779)

Consider the two points M(tan(A+t):tan(B+t):tan(C+t)) and N(tan(A-t):tan(B-t):tan(C-t)) in barycentric coordinates where t is any real number. It is clear that these points are collinear with H. As t varies, their locus is the cubic K267.

When the two points M, N above are expressed in trilinear coordinates, the locus is K457. See also Table 38.

Note that the barycentric product X(1) x K267 is K457 and the barycentric product X(6) x K267 is K1057 which is at the time the isogonal transform of K267.

K267 is an isotomic nodal nK with root R = X(14570) = X5 x X99, a point on the line X(523)-X(1634), and node G which is always an isolated point on the curve. Any line through H, M, N meets the Kiepert and Jerabek hyperbolas again at k and j respectively. M and N are harmonically conjugated with respect to k and j. In other words, the circles with diameters MN and jk are orthogonal.

H is a flex on the curve and the inflexional tangent passes through X(51), the centroid of the orthic triangle. Two other inflexion points F1, F2 are studied below.


The pivotal conic has center X(338) and touches K267 at X(69), tF1, tF2. The isotomic conjugates of these points are X(4), F1, F2 which are the real inflexion points of K267.

Note that the line X(4)X(94) contains tF1, tF2 hence F1, F2 lie on the circumconic passing through X(69), X(323), X(340).

Recall that this pivotal conic is inscribed in the antimedial triangle. It is the anticomplement of the inconic with perspector X(249).

The contact conic is the circumconic passing through X(69), X(1994), tF1, tF2 and E = tX562, this latter point lying on the pivotal conic.

The isotomic transform of the contact conic is the (blue) line X(4)X(93) passing through F1, F2 thus giving a construction of F1, F2.

Now in ETC :

F1, F2 are X(19778), X(19779)

tF1, tF2 are X(16770), X(16771)