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(y - z) [a^4 y z + b^2 c^2 x (2x + y + z)] = 0

X(2), X(6), X(61), X(62), X(251)

(blue) Ix-anticevian points, see table 23, on the hyperbola (H) passing through X{5, 6, 52, 195, 265, 382, 2574, 2575}

foci of the K-ellipse (inellipse with center K when the triangle ABC is acutangle)

other points below

Geometric properties :

K755 is the isogonal transform of K754. These two cubics belong to the Brocard-Grebe pencil where they are denoted by Γ(2) and Γ(-2) respectively. See Table 52.

K755 meets the line at infinity and the circumcircle at the same points as K655 = pK(X6, X141) and pK(X6, X3629) respectively.

K755 is spK(X141, X6) in CL055.

A generalization connected with polar conics

Let P be a point on the line at infinity and let (Kp) be the isogonal circular pivotal cubic pK(X6, P). The orthic line of (Kp) is the line OP hence the polar conic of O in (Kp) is a rectangular hyperbola. When P traverses the line at infinity, these polar conics belong to a same pencil hence they pass through four fixed points, independent of P, which are the vertices of an orthocentric quadrilateral with diagonal triangle T.

When P is the infinite point of an altitude of ABC, say AH, (Kp) is a focal cubic with focus A and the polar conic of O in (Kp) is a simple rectangular hyperbola (Ha) passing through :

• A, with tangent the symmedian AK,

• the reflections of A in B, C, Ga = A-vertex of the antimedial triangle which is the center of (Ha),

• the infinite points of the A-bisectors of ABC.


A net of cubics

Now, let Q = u : v : w be a point and let K(Q) be the cubic with equation (v + w) x (Ha) which obviously contains A, B, C and the four fixed points above.

It is easy to verify that K(Q) is in fact spK(cQ, Q) where cQ is the complement of Q. See CL055.

It follows that K(Q) must pass through :

• the isogonal conjugate gcQ of cQ,

• the reflection scQ of cQ in Q,

• the infinite points of pK(X6, cQ),

• the points on (O) of pK(X6, scQ),

• the foci of the inconic with center Q (when Q is not an in/excenter).

• the traces on the sidelines of ABC of the reflections (Da), (Db), (Dc) in Q of the corresponding cevian lines of cQ. This gives another form of the equation of K(Q) which is a^2 y z (Da).

For example, with Q = X2, X4, X6 we obtain the cubics K002, K525, K755 respectively. See CL066 for other examples and further properties.


A related conjugation and a remarkable triangle

The polar lines of a point M in the pencil of rectangular hyperbolas above concur at M* which is actually the isogonal conjugate of M with respect to the diagonal triangle T. The four points are therefore the in/excenters of T.

With M = x : y : z , we have M* = a^2 y z + b^2 z x + c^2 x y – a^2 (x + y + z)^2 + b^2 x (x + y – z) + c^2 x (x – y + z) : : .

{M, M*} = {X(i), X(j)} for these {i, j} : {1, 1046}, {2, 3629}, {4, 185}, {6, 251}, {30, 146}, {61, 62}, {145, 8256}, {147, 511}, {148, 512}, {149, 513}, {150, 514}, {151, 515}, {152, 516}, {153, 517}, {193, 1368}, {194, 3491}, {523, 3448}, {524, 14360}, {525, 13219}, {1503, 12384}, {1510, 11671}, {5668, 5669}, (Peter Moses).

Note that points at infinity are transformed into points on C(X4, 2R).


In general, this conjugation transforms K(Q) into a sextic passing through the images of A, B, C which are the vertices A', B', C' of a remarkable triangle with coordinates A' = a^2-b^2-c^2 : b^2 : c^2, B' = a^2 : -a^2+b^2-c^2 : c^2, C' = a^2 : b^2 : -a^2-b^2+c^2, very similar to the second Brocard triangle A2B2C2. If KaKbKc is the cevian triangle of K = X(6) then A' is the harmonic conjugate of A2 in A and Ka or, equivalently, the inverse of A2 in the circle with diameter AKa.

These points A', B', C' also lie on K534 which is a pK with pivot X(69) in A'B'C'.


K755 is the only cubic of the net that passes through A', B', C' hence the only cubic invariant under the conjugation.

More precisely, K755 is the isogonal pK with pivot X(6) with respect to T.

In the figure, the triangle T is labelled Q1Q2Q3 and its in/excenters are R0, R1, R2, R3.


Remark 1 : these four latter points also lie on two simple rectangular hyperbolas (H1) and (H2) :

(H1) passes through X3, X6, X382, X2574, X2575 hence it is homothetic to the Jerabek hyperbola. It is the polar conic of O in the Neuberg cubic K001.

(H2) passes through X2, X8, X3146, X3413, X3414 hence it is homothetic to the Kiepert hyperbola. It is the polar conic of O in pK (X6, X511).

Remark 2 :

The vertices Q1, Q2, Q3 of T lie on C (X4, 2R) and also on two simple rectangular hyperbolas (H3) and (H4) :

(H3) passes through X146, X185, X193, X2574, X2575, also homothetic to the Jerabek hyperbola.

(H4) passes through X147, X185, X194, X3413, X3414, also homothetic to the Kiepert hyperbola.

It follows that X185 must be the orthocenter of T hence the isogonal conjugate of X4 with respect to T. See below for other remarkable centers of T.

Further properties of T (with contributions by Peter Moses) :

• T is also inscribed in K417 which is an isogonal nK in this triangle. It is the locus of foci of conics inscribed in T whose center lie on the Brocard axis of ABC.

• the sidelines of T are parallel to the asymptotes of nK0(X6, X1194).

• this same nK0 meets the circumcircle at A, B, C and three other points whose anticomplements are precisely the vertices of T.

• The in/excenters R0, R1, R2, R3 of T live on K002, K525, K755, K915, K916, K918, Q060, Q069.

• X(i) of T is X(j) of ABC for these {i, j} : {2, 51}, {3, 4}, {4, 185}, {5, 389}, {20, 11381}, {30, 6000}, {140, 10110}, {381, 5890}, {382, 6241}, {523, 520}, {526, 6086}, {546, 13382}, {550, 13474}, {1511, 10745}, {1656, 3567}, {1657, 12290}, {3526, 9781}, {3534, 11455}.

It follows that the Euler line of T is the line X(4), X(51), X(185), X(389), X(1075), X(1093), X(1896), X(1899), X(2052), X(3168), etc.

• X(i) of the excentral triangle of T is X(j) of ABC for these {i, j} : {5, 4}, {51, 51}, {52, 185}, {143, 389}, {1154, 6000}, {1510, 520}, {5562, 11381}, {10095, 10110}, {11591, 13474}.

Properties of A'B'C' (Peter Moses)

A'B'C' is perspective to :

ABC at X(6)

tangential (TCCT 6.5)

second Brocard (CTC)

symmedial (MathWorld)

inner Grebe (see ETC X(1160))

outer Grebe (see ETC X(1161))


second Ehrmann (see ETC (8537))

fifth Brocard (see ETC X(32))

second orthosymmedial (see ETC X(6792))

1st Kenmotu diagonal (see X(31))

2nd Kenmotu diagonal (see X(31))

inner tri-equilateral (see X(10631))

outer tri-equilateral (see X(10631))

anti-Conway (see X(11363))

medial of orthic / anti 2nd Conway (see X(11363))

anti-5th Brocard


the cevian (resp. anticevian) triangle of every point on K531 (resp. K538).

A'B'C' is orthologic to :

second Neuberg (MathWorld)

first Ehrmann (see ETC (8537))