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X(3), X(4), X(64), X(1676), X(1677), X(6759)

vertices of the CircumNormal triangle

Z = X(6759) = X(3)X(64) /\ X(4)X(54) /\ X(5)X(182), etc

vertices of the pedal triangle of Z

isogonal conjugates of Ua, Ub, Uc mentioned in the Neuberg cubic page. See also table 16 and table 18.

X3-OAP points, see Table 53

other points and properties below

K405 is a CircumNormal cubic and actually the locus of isopivots of CircumNormal pKs. The loci of their poles and pivots are K378 and K361 respectively. See Table 25.

K405 is the isogonal transform of K080 = KO++ and spK(X20, X550) in CL055. It follows that K405 has three real asymptotes parallel to the altitudes, hence parallel to those of the Darboux cubic. See other analogous cubics in Table 58.

K405 is also the cubic Dk with k = 1/2. See Table 33. The related cubic Lk is K026.

K405 is a member of the pencil of cubics generated by the McCay cubic K003 and the Lemoine cubic K009. It is the only cubic – apart K009 – meeting the sidelines of ABC at the vertices of a pedal triangle, that of Z described above. The polar conic of Z contains the vertices Ta, Tb, Tc of the CircumNormal triangle and H. It follows that the tangents at these points pass through Z and that Z is the tangential of H.

K405 is the isogonal pK with respect to the CircumNormal triangle with pivot X(4) and isopivot X(6759). Hence, K405 passes through Ha = TbTc /\ X(4)Ta, Hb and Hc likewise.

See also Table 54 where K361 is mentioned in one line and three columns showing that it belongs to four pencils of cubics generated by :

• K080 and K187, line Q = X550,

• K003 and the union of the Euler line with the circumcircle, column P = [X3], already mentioned above,

• K004 and the union of the line at infinity and the Jerabek hyperbola, column P = X20,

• K006 and K026, column P = S.

O, with tangent the Euler line.

H, with tangent through Z and X(54), the Kosnita point.

X(64), the isogonal conjugate of the de Longchamps point X(20).

X(1676), X(1677) the similicenters of the nine point and first Lemoine circles. The line through these two points contains Z.

Ka, Kb, Kc : vertices of the tangential triangle. The tangents at these points concur at X(160).

Oa, Ob, Oc : isogonal conjugates of the reflections of H in A, B, C. These are the last points on the cevians of O.

Ta, Tb, Tc : vertices of the CircumNormal triangle with tangents passing through Z = X(6759).

T'a, T'b, T'c : images of Ta, Tb, Tc under the homothety with center O, ratio –2. These are the vertices of an equilateral triangle inscribed in K405 and C(O,2R).

Ua*, Ub*, Uc* : the other common points of K405 and C(O, 2R). These points lie on the Neuberg cubic. More details in the Neuberg cubic page.

The "last" points on the cevian lines of X(64) are the isogonal transforms (in the CircumNormal triangle) of Ka, Kb, Kc.

(with contributions by Angel Montesdeoca, 2017-05-06)

Consider a point P with pedal triangle PaPbPc. Let OaObOc be the triangle with vertices the projections of O = X(3) on the lines PPa, PPb, PPc respectively.

1. ABC and OaObOc are perspective (at Q) if and only if P lies on K405. The locus of the perspector Q is K080, the isogonal transform of K405.

2. For any P, ABC and OaObOc are orthologic (at P and O1) and parallelogic (at O and P1). O1 is the isogonal conjugate of the infinite point of the line OP and P1 is its antipode on (O). It follows that ABC and OaObOc are inversely similar for any P.

3. Hence, when P lies on K405, ABC and OaObOc are bilogic and their center of similitude Ω lies on K361.

4. Let f be the mapping that sends P onto Ω = f(P).

A few pairs {P, Ω} are : {X3, X3}, {X4, X54}, {X64, X4}, {X1676, X1342}, {X1677, X1343}.

f is a quadratic (non involutary) mapping with three singular points namely Ta, Tb, Tc and four fixed points namely O, T'a, T'b, T'c as mentioned above. It follows that f transforms any curve of degree n not passing through Ta, Tb, Tc into a curve of degree 2n passing through Ta, Tb, Tc. Note that f maps the line at infinity onto the circumcircle (O) and, in particular f(X30) = X(110), f(X511) = X(99).

For example, f transforms the Euler line into the Jerabek hyperbola of the CircumNormal triangle which is also the polar conic of O in K361. This hyperbola passes through Ta, Tb, Tc, X(3), X(54), X(110), X(182), X(1147), X(1385), X(2574), X(2575), X(6759), X(8717), X(8718), X(8723), X(8907), X(9932), X(11935), X(12584), X(12893), etc.

Also, f transforms the Brocard axis into the rectangular hyperbola passing through Ta, Tb, Tc, X(3), X(99), X(376), X(3413), X(3414). More generally, f transforms any line through X(3) into a rectangular hyperbola passing through Ta, Tb, Tc, X(3). See Table 25 for other examples.

Let P be a fixed point. The locus of M such that P, M, f(M) are collinear is a cubic K(P) passing through P, Q = f^–1(P) and X(3), Ta, Tb, Tc, T'a, T'b, T'c since these latter seven points are the singular and fixed points of f. These cubics are therefore in a same net.

Similarly, the locus of M such that P, M, f^–1(M) are collinear is a cubic K'(P) passing through P, Q = f(P) and X(3), Ta, Tb, Tc. The tangents at Ta, Tb, Tc concur at X(3). These cubics are therefore also in a same net.

K(P) and K'(P) meet the circumcircle (O) at the same six points namely Ta, Tb, Tc and three other points Sa, Sb, Sc . Their three remaining common points are P and X(3) counted twice since the common tangent at X(3) passes through P.

When P = X(4), Sa, Sb, Sc are the vertices of ABC and when P= X(3) they are the vertices of the CircumTangential triangle, see below.

In all other cases, let E be the isogonal conjugate of the infinite point of the line X(3)P. Sa, Sb, Sc are the common points (apart E) of (O) and the rectangular hyperbola H(P) passing through X(3), P, f(P), f^–1(P), E and the infinite points of the rectangular hyperbola which is the isogonal transform of the line X(3)P.

K(P) and K'(P) meet the line at infinity at the same points as the isogonal pKs with pivots the reflection S of P in X(3) and the midpoint S' of X(3), P respectively.

Special cases :

• K(X3) = K'(X3) decomposes into the altitudes of the CircumNormal triangle. This is the only equilateral cubic in each net.

• K(X4) = K405 and K'(X4) = K361 are the only circum-cubics in each net.

• K(P) and K'(P) are circular if and only if P lies on the line at infinity.