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X(187), X(538), X(671), X(729), X(39162), X(39163), X(39164), X(39165), X(39641), X(39642), X(47646), X(51492), X(51493), X(51494)

X(39162), X(39163), X(39164), X(39165) are the foci of the inscribed Steiner ellipse, see also Table 48.

Ω1, Ω2 : Brocard points

X(39641), X(39642) are the imaginary foci of the Brocard ellipse

other points below

K248 is an isogonal circular focal nK with root X(512), singular focus X(729).

It is the locus of foci of inscribed conics centered on the line X(2)X(39) or equivalently tangent to the trilinear polar of X(512). This line meets the sidelines of ABC at A', B', C' on the cubic.

Hence, the cubic contains the foci F1, F2 of the inscribed Steiner ellipse (S) and the Brocard points, foci of the Brocard ellipse (B). The trilinear polar of X(512) is obviously tangent to these two ellipses at X(1084) and X(3124) respectively.

K248 also contains the imaginary foci of these two ellipses.

K248 is also spK(X538, X2) as in CL055 and SpK(X187, L) as in CL056 where L is the Brocard axis.

Points on K248

Y* = X(47646), on the lines {98, 39291}, {99, 187}, {671, 729}, {880, 5026}, {2782, 33757}, {3225, 10754}, {3972, 43765}, {4027, 17941} and on K705. Y* is the common tangential of the Brocard points.

Y = X(51494) = isogonal conjugate of X(47646), on the lines {39, 512, Ω1, Ω2}, {187, 729} and on K353.

X = intersection of K248 with its real asymptote and common tangential of X(538) and X(729).

X* = isogonal conjugate of X, on the line X(538), X(729).

Z = third point on the line X(187), X(671).

Z* = isogonal conjugate of Z, common tangential of X(187) and X(671).

X(51492), X(51493) are the third points on the axes of the inscribed Steiner ellipse. These points lie on the line X(2), X(99) and on K553, a nodal circum-cubic that contains the Brocard points.

K248 contains the isogonal conjugates X(51492)*, X(51493)* of X(51492), X(51493). These points lie on the line X(729), X(1084) and on the circum-conic with perspector X(888) passing through X(2), X(512), X(538), X(3231). It follows that they are the common tangentials of the pairs of foci of the inscribed Steiner ellipse that lie on a same axis.

The two cubics K248, K553 have two remaining common points Q1, Q2 which lie on the line X(671), X(3124) and on the circum-conic with perspector X(351), passing through X(6), X(187), X(249), X(512), X(524), X(598), X(843), etc.

This conic is the isogonal transform of the line X(2), X(99) and the line X(671), X(3124) is the isogonal transform the circum-conic passing through X(187), X(4590). Hence Q1, Q2 are the isogonal conjugates of two points B1, B2 which are actually the last common points of K248 on the Brocard circle.