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X(2) X(6), X(194), X(732), X(3413), X(3414), X(4027), X(6776), X(8787), X(9755), X(39641), X(39642)

X(39641), X(39642) are the imaginary foci of the Brocard inellipse, on the Brocard axis and the Kiepert hyperbola

vertices of the symmedial triangle

Geometric properties :

Let W be a point on the Brocard axis such that OW = t OK (vectors) and let W* be its isogonal conjugate hence on the Kiepert hyperbola (K).

S is the second point of (K) on the line (L) passing through W* and X(5). At last, T is defined by ST = 2t SK (vectors) where K = X(6).

Remark : if S' is the midpoint of ST and W' the reflection of O in W then the lines OS, WS', TW' are parallel.

When t varies, T lies on the nodal cubic K1113 with node X(194).

Two asymptotes are parallel to those of (K) and meet at X = 2 a^8+3 a^6 b^2-3 a^4 b^4+3 a^2 b^6+3 a^6 c^2-12 a^4 b^2 c^2+2 a^2 b^4 c^2-2 b^6 c^2-3 a^4 c^4+2 a^2 b^2 c^4+4 b^4 c^4+3 a^2 c^6-2 b^2 c^6 : : , on the lines {39, 98}, {99, 5008}, {115, 5066}, {148, 598}, {620, 5215}, etc. X = X(33694) in ETC (2019-07-30).

Other centers on the cubic (1st coordinate) :

P1 = 2 a^3+2 a^2 b+a b^2+2 a^2 c+2 a b c+b^2 c+a c^2+b c^2

P2 = (a^2-2 b^2-2 c^2) (8 a^4+a^2 b^2+2 b^4+a^2 c^2-5 b^2 c^2+2 c^4)

P3 = (3 a^2-b^2-c^2) (7 a^4-6 a^2 b^2+3 b^4-6 a^2 c^2-10 b^2 c^2+3 c^4)

P4 = (3 a^4+a^2 b^2+a^2 c^2-2 b^2 c^2) (3 a^4-4 a^2 b^2-3 b^4-4 a^2 c^2-10 b^2 c^2-3 c^4)

P5 = (a^4+2 a^2 b^2+2 a^2 c^2+b^2 c^2) (3 a^4+6 a^2 b^2+2 b^4+6 a^2 c^2+5 b^2 c^2+2 c^4)

P6 = (2 a^4-a^2 b^2-a^2 c^2-3 b^2 c^2) (8 a^4+a^2 b^2+2 b^4+a^2 c^2-5 b^2 c^2+2 c^4)

P7 = (b+c) (2 a^3+2 a^2 b+a b^2+2 a^2 c+2 a b c+b^2 c+a c^2+b c^2) (a^3 b+a^2 b^2+a^3 c+a^2 b c+a^2 c^2+b^2 c^2)

P8 = (a^4-2 a^2 b^2-2 a^2 c^2-3 b^2 c^2) (5 a^4-2 a^2 b^2+2 b^4-2 a^2 c^2-5 b^2 c^2+2 c^4)

P9 = (5 a^4-a^2 b^2-a^2 c^2-6 b^2 c^2) (35 a^4+4 a^2 b^2+5 b^4+4 a^2 c^2-26 b^2 c^2+5 c^4)

P10 = (a^6-2 a^2 b^2 c^2-b^4 c^2-b^2 c^4) (a^8+2 a^6 b^2+a^4 b^4+a^2 b^6+2 a^6 c^2+a^4 b^2 c^2+a^4 c^4-b^4 c^4+a^2 c^6)

P11 = (3 a^4-a^2 b^2-a^2 c^2-4 b^2 c^2) (15 a^4+2 a^2 b^2+3 b^4+2 a^2 c^2-10 b^2 c^2+3 c^4)

P12 = (b^2-b c+c^2) (b^2+b c+c^2) (a^8+2 a^6 b^2+a^4 b^4+a^2 b^6+2 a^6 c^2+a^4 b^2 c^2+a^4 c^4-b^4 c^4+a^2 c^6)

These points are now X(33682) up to X(33693) in ETC (2019-07-30).

A related parabola

When t varies, the line WT envelopes a parabola (P) passing through X(20), X(182), X(6776), X(29012) at infinity.

The tangents to (P) at these points are the Euler line, the Brocard axis, the line HK (also tangent to K1113), the line at infinity.

The focus F = X(33695) lies on the lines {5, 83}, {39, 112}, {185, 575}, {9479, 32467}, {11643, 14908}, {13335, 22467} and on the circumcircle (C) of X(3)X(4)X(6). F is the reflection of X(827) in X(14675).

The directrix (D) is the line passing through X(54), X(826) at infinity, the orthocenter X(879) of X(3)X(4)X(6).

(P) is inscribed in triangle X(3)X(4)X(6) and its perspector is X(25406), a point on the Steiner ellipse of X(3)X(4)X(6) and on the lines {2, 154}, {3, 69}, {4, 83}, {5, 12017}, {6, 20}, etc.

The circumcenter Ω = X(33752) of X(3)X(4)X(6) lies on {5, 523}, {32, 2485}, {114, 132}, etc, SEARCH = 9.73953577571382.