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X(3), X(30), X(110), X(1157), X(2080), X(6760), X(8724)

inverses of X(1147), X(1385)

vertices of the CircumNormal triangle

X = X(14933) = X(30)X(74) /\ X(3)X(526) on the asymptote

Y = X(14934) = X(30)X(110) /\ X(3)X(523)

reflections of A, B, C in X(5) or reflections of X(3) in the sidelines of ABC i.e. centers of the Johnson circles

A variable line L passing through X(110) meets the Euler line at S and the circle C(S, X3) at two points S1, S2 of the Neuberg strophoid K725.

The Neuberg cubic K001 and K725 have the same focus namely X(110), the same real asymptote namely the parallel at X(74) to the Euler line and therefore the same orthic line which is the Euler line. Hence, K001 and K725 meet at nine identified points : X(3), X(30), the circular points at infinity (each counted twice) and X(1157) which is the tangential of X(3) in K001. The nodal tangents at X(3) are parallel to the asymptotes of the Jerabek hyperbola.

The locus of the midpoints M1, M2 of X3-S1, X3-S2 is the Stammler strophoid K038 and the perpendicular bisectors of X3-S1, X3-S2 are tangent to the Kiepert parabola. Note that the line X(3)S1 meets K001 again at two points whose midpoint is M1.

K725 is :

• the homothetic of K038 under h(X3, 2).

• the reflection of K025 about X(5).

• the reflection of K905 about X(3).

• (equivalently) the pedal curve of X(3) in the parabola with focus X(399), directrix the Euler line. This parabola is the homothetic of the Kiepert parabola under h(X3, 2).

• (equivalently) the locus of the reflections of X(3) in the tangents to the Kiepert parabola.

• the locus of contacts of tangents drawn through X(110) to the circles passing through X(3) and tangent at X(3) to the Euler line. See a generalization below.

• the inverse in the circumcircle of the remarkable rectangular hyperbola (H) described in Table 25 which contains X(3), X(54), X(110), X(182), X(1147), X(1385), X(2574), X(2575) and the vertices of the CircumNormal triangle N1N2N3. It follows that K725 is a CircumNormal cubic.

• a member of the pencil of cubics generated by K001 and the decomposed cubic which is the union of the line at infinity (counted twice) and the tangent at X(3) to K001 (the line passing through X(3), X(54), etc).

***

The polar conic of X(110) is the circle passing through X(3), X(110), X(477) whose center is the contact of the perpendicular bisector of X(3)X(110) with the Kiepert parabola.

K725 meets the sidelines of N1N2N3 again at three collinear points F1, F2, F3 which lie on the perpendicular bisector (L) of X(3)X(110).

These points are the inverses of the common points E1, E2, E3 of (H) and the circle (C) = C(X110, X3), excluding X(3). The triangle E1E2E3 is obviously equilateral.

The perpendicular bisectors of X3-Fi are three tangents to the Kiepert parabola.

The polar conic of X(74) splits into the Euler line and the line (L) hence the tangents at F1, F2, F3 to K725 concur at X(74).

K725 is an isogonal cK with respect to the CircumNormal triangle.

For any point M on K725, the CircumNormal isogonal conjugate M* of M lies on K725 and the line (T) passing through M, M* is tangent at N to the parabola (P) which is the homothetic of the Kiepert parabola under h(X3, 2). Hence its directrix is the Euler line and its focus is X(399).

The root of K725 is X(523) since its trilinear polar with respect to the CircumNormal triangle is the perpendicular bisector (∆) of X(3)X(110).

See Special Isocubics, chapter 8, for further properties concerning cubics cK.

Generalization

Let S be a point on the Euler line of ABC and (C) a variable circle passing through X(3) and S. If S = X(3), (C) is meant to be tangent at X(3) as said above.

The locus of contacts of tangents drawn through X(110) to (C) is a focal cubic F(S) which is also the locus of the common points of (C) with the polar line of X(110) in (C).

For any point S,

• the singular focus of F(S) is X(110) and its polar conic is the circle passing through X(110), X(3) and S.

• the real asymptote is that of the Neuberg cubic K001, namely the line X(30), X(74).

• the orthic line is the Euler line.

• the tangent at X(3) passes through X(110), unless S = X(3) in which case it is the node of the strophoid K725.

• F(S) meets the circumcircle at X(110), the circular points at infinity and the same points as pK(X6, S) which are not the vertices of ABC. These points Q1, Q2, Q3 are not necessarily all real and they also lie on the rectangular hyperbola passing through S and the midpoints of the segments with endpoints S and the in/excenters of ABC. Recall that the orthocenter of Q1Q2Q3 is S.

• Hence F(S) and pK(X6, S) meet at X(3), S, Q1, Q2, Q3 and two pairs of points P1, P2 (real) and P3, P4 (imaginary conjugate) with same midpoint cS, the complement of S. These points lie on two perpendicular lines secant at cS and parallel to the asymptotes of the rectangular circum hyperbola passing through Y on the Euler line, the image of X(2) under the homothety with center X(3), ratio (3 - t)^2 / (12 - 8t) where t is the abscissa of S in X(3), X(2).

When S traverses the Euler line, the locus of these four latter points is K800.

• F(S) meets the sidelines of Q1Q2Q3 again at three collinear points lying on the perpendicular bisector M(S) of X(110)-S.

• F(S) is an isogonal nK with respect to Q1Q2Q3. It is therefore the locus of foci of conics inscribed in Q1Q2Q3 with center on the Euler line.

• F(S) meets the isogonal pK with pivot S with respect to Q1Q2Q3 at X(3), S, Q1, Q2, Q3 and four other points two by two symmetric about S and lying on the parallels at S to the asymptotes of the rectangular hyperbola passing through A, B, C, X(4), S. If S = X(4), see K799, the hyperbola is tangent at X(4) to the Euler line.

• F(S) meets the perpendicular bisectors of ABC at X(3) and six other points two by two symmetric about the sidelines of ABC.

***

All these cubics are in a same pencil which contains :

• the decomposed cubic which is the union of the line at infinity (twice) and the line X(3)X(110), obtained when S = X(30),

• the nodal cubic K725, obtained when S = X(3),

• the cubic K463, obtained when S = X(2),

• the reflection of K187 about X(3), obtained when S = X(20),

• the cubic K798, obtained when S = X(140),

• the cubic K799, obtained when S = X(4),

• the cubic K853, obtained when S = X(2475), the only one passing through X(1),

• an axial cubic with axis the perpendicular at X(110) to the Euler line. S is complicated and unlisted in ETC.

***

Each cubic F(S) belongs to a pencil generated by the Neuberg cubic and the union of the line at infinity (counted twice) with a line L(S) passing through X(3). L(S) is the trilinear polar of the X(110)-isoconjugate of cS, the complement of S.