X(4), X(30), X(265), X(316), X(671), X(1263), X(1300), X(5080), X(5134), X(5203), X(5523), X(5962), X(10152), X(11604), X(11605), X(11703), X(13495)
foci of the Steiner circum-ellipse
Two points M and N have equal length cevians (i.e. AMa = ANa, etc) if and only if equivalently :
Ho is the isogonal transform (with respect to the orthic triangle) of the Euler line of ABC. It contains the vertices of the cevian triangles of H = X(4) and X(648) = trilinear pole of the Euler line and the triangle centers X(4), X(6), X(52), X(113), X(155), X(185), X(193), X(1162), X(1163), X(1829), X(1839), X(1843), X(1858), X(1986). Its center is X(1112). Its equation is : (b^2 - c^2)(SA^2 x^2 + SB SC y z) + cyclic = 0. See also Clark Kimberling, Conics Associated with a Cevian Nest, Forum Geometricorum, vol. 1 (2001), 141 -- 150.
K025 contains the foci of the Steiner circum-ellipse since the six cevians of these two points all have the same length.
The tangents to the strophoid at H are parallel to the asymptotes of the Jerabek hyperbola : they are the Steiner lines of the points where the Euler line meets the circumcircle. The singular focus is F = X(265) isogonal conjugate of the inversive image of H in the circumcircle. The real asymptote is parallel to the Euler line and pass through X(110), X(477) and the reflection of F in H. The bisectors of (FM,FN) are parallel and perpendicular to the Euler line.
A construction :
A variable line L passing through the singular focus F = X(265) meets the Euler line at P. The circle with center P passing through the orthocenter H of ABC meets the line L at two points on the cubic.
Locus properties :
A generalization of property 5 can be found in Hyacinthos messages #1811, 1812, 1816, 1821, 1822, 1823, 1831, 1833 where H is replaced by a point R and the altitudes by the cevian lines of R. In general, we obtain a circular nodal cubic with node R and the nodal tangents are always perpendicular. When R is an in/excenter, the locus is the whole plane. When R lies on the circumcircle, the cubic decomposes into the circumcircle and a line through O. When R lies at infinity, the cubic decomposes into the line at infinity and a circum-rectangular hyperbola. When R = O, we obtain the Jerabek strophoid K039.
See another generalization and other related cubics in Table 43.
The reflection of K025 about X(5) is K725.
Generalization of property 1
Let (L) be a line through the orthocenter H of ABC which can be seen as the Steiner line of a point N on the circumcircle (O).
The antigonal transform (K) of (L) is a circular nodal circum-cubic with node H.
M is the isogonal conjugate of the infinite point of (L) and the antipode of N on (O).
(C) is the bicevian conic passing through H and the vertices of the orthic triangle which is tangent at H to (L). It is the orthoassociate of (K) i.e. its inverse in the polar circle.
E is the complement of M, a point on the nine point circle and on (C).
F is the singular focus of (K). It is the image of M under the translation that maps O onto H, hence F is a point on the circle C(H, R).
S is the last point of (K) on (O). It is the orthoassociate of E.
The nodal tangents at H to (K) are parallel to the asymptotes of (C).
The real asymptote of (K) is the parallel at N to (L).
The following table gives a selection of remarkable cubics (K). (L) is the line passing through H and P.