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X(2), X(30), X(98), X(325), X(1494), X(3505), X(14941), X(14944), X(16070), X(16071)
infinte points of the Steiner ellipses
X = X(16075) = reflection of X(1494) about X(1650)
its isotomic conjugate tX = X(16076) = reflection of X(1494) about Y = X(16077) = isotomic conjugate of X(9033)
U, V, W on the trilinear polar of X(525)
Geometric properties :
K953 is the locus of M such that the midpoint of M and its isotomic conjugate tM lies on the Euler line. When "isotomic" is replaced with "isogonal", the locus is K187.
K953 is also the antitomic transform of the Euler line. See ETC, preamble of X(14941). More precisely, if M and M' are two points on the Euler line conjugated in the Steiner ellipse then their antitomic transforms are two isotomic conjugate points on K953. Further details below.
Recall that K025 is the antigonal transform of the Euler line.
K953 is a conico-pivotal cubic with pivotal conic (P), the parabola with focus X(13219) and directrix the line passing through X(20), X(64). Obviouly, the nodal tangents at X(2) to K953 are tangent to (P).
The polar conic of X(1494) is an ellipse passing through X(2), X(1494), Y which is homothetic to the Steiner ellipses. Note that Y also lies on the Steiner ellipse and on the contact conic (C).
The real asymptote of K953 is parallel to the Euler line at X(648) and meets the cubic again at X.
The two imaginary asymptotes meet at X(1494) and the tangent to K953 at X(1494) passes through X.
K953 is in many ways similar to a focal cubic where the circular points at infinity are replaced with the infinte points of the Steiner ellipses, in which case X(1494) is analogous to the singular focus.
Let us consider two conjugated diameters in the Steiner ellipse with infinite points I1, I2. The barycentric quotients N1, N2 of X(525) and I1, I2 are two points on the circum-conic with perspector X(525), the isotomic transform of the Euler line.
The X(2)-Hirst conjugates M1, M2 of N1, N2 lie on K953 and are collinear with X(1494). Their isotomic conjugates tM1, tM2 are also two points of K953 collinear with X(1494).
• the lines M1-tM1 and M2-tM2 – which are tangent to the pivotal conic (P) – meet at N on the line X(2)X(525),
• the lines M1-tM2 and M2-tM1 are parallel to the Euler line.