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X(4), X(110), X(1113), X(1114)

infinite points of the altitudes

traces of the Simson line of X(110)

K839 is an example of nK related with Table 58 where a generalization can be found. It is the sister of K316 = pK(X6, X110). See also K842.

K839 is a curious case of isogonal pivotal cubic (with respect to the triangle T = X110-X1113-X1114) since its pivot H lies on one of its sidelines.

Recall that X(110) is the unique finite point which is invariant in the transformation P –> Z described in Table 58. It follows that X(110) is a flex on the cubic with inflexional tangent passing through X(924). The harmonic polar line of X(110) is the Euler line and then the tangents at X(4), X(1113), X(1114) pass through X(110).

K839 meets :

• the sidelines of ABC at Za, Zb, Zc on the Simson line of X(110). In other words, Za, Zb, Zc are the feet of the perpendiculars drawn from X(110) to the sidelines of ABC.

• the circumcircle again at X(1113), X(1114) on the Euler line i.e. on the Steiner line of X(110).

Naturally, the isogonal conjugates (with respect to T) of these five points also lie on the cubic.

K839 meets K004 at three points at infinity, A, B, C, H and two other (not always real) points on the line (L) passing through X(110), X(1498) and on the rectangular circum-hyperbola (H) passing through X(2071), X(2693).

K839 is the isogonal transform (with respect to ABC) of K069.