K003


Home page \| Catalogue \| Classes \| Tables \| Glossary \| Notations \| Links \| Bibliography \| Thanks \| Downloads \| Related Curves






	X(1), X(3), X(4), X(1075), X(1745), X(3362), X(13855) Ia, Ib, Ic vertices of excentral triangle vertices of the circumnormal triangle foci of the inconic with center O, perspector X(69) imaginary foci of the Brocard ellipse i.e. common points of the Brocard axis and the Kiepert hyperbola anti-points, see Table 77 more points and details below

The McCay cubic is the isogonal pK with pivot O = X(3). See Table 27. It is sometimes called Griffiths cubic. It is a member of the classes CL006, CL009, CL021, CL024 of cubics. See also Table 63. It is the only isogonal pK60+ and the three asymptotes concur at G. These asymptotes meet the cubic at three finite points which lie on the line homothetic of the Brocard line OK under h(G,2/3). The McCay cubic is an example of stelloid. See also K024, Table 22 and Table 51.

Since K003 is an equilateral cubic, the polar conic of any point of the plane is a rectangular hyperbola. The locus of centers of the polar conics of the points on the McCay cubic is Q048, the McCay sextic. In other words, the Psi transform of K003 is Q048. More on polar conics below.

The isotomic transform of K003 is pK(X76, X69). The symbolic substitution SS{a -> √a} transforms K003 into K977.

The CircumTangential-isogonal transform of K003 is K1267.

Locus properties (see also Table 7)

Locus of point P such that the polar lines (in the circumcircle) of P and its isogonal conjugate P* are parallel. Equivalently, locus of P such that P, P*, iP (inverse of P in the circumcircle) are collinear. Compare this with K105, property 1.
Locus of point P such that the polar lines (in the inscribed ellipse with center O, perspector X(69)) of P and its isogonal conjugate P* are parallel.
Denote by PaPbPc the pedal triangle of point P and by T1, T2, T3, T4, T5, T6 the areas of triangles PBPa, PPaC, PCPb, PPbA, PAPc, PPcB respectively. The locus of point P such that T1*T3+T3*T5+T5*T1 = T2*T4+T4*T6+T6*T2 is the McCay cubic. See also K004 (property 3) and Q082.
Locus of point P such that the sum of line angles (BC,AP)+(CA,BP)+(AB,CP)=pi/2 (mod. pi). Compare this with K024, property 6. See also a generalization in Table 22.
Locus of point P whose pedal circle is tangent to the nine point circle. See a generalization in table 22. When the nine point circle is replaced with the incircle we obtain the isogonal sextic Q104. When pedal circle is replaced with cevian circle, we obtain a sextic which is the isotomic transform of the anticomplement of K102.
Locus of point P whose pedal and circumcevian triangles are perspective and even homothetic. The perspector is a point of the Lemoine cubic. See also CL024.
Locus of point P such that the antipedal triangle of P and the circumcevian triangle of its isogonal conjugate are perspective (together with the line at infinity and the circumcircle). These triangles are in fact homothetic.
Denote by PaPbPc the circumcevian triangle of point P and by QaQbQc the triangle bounded by the Simson lines of Pa, Pb, Pc. PaPbPc and QaQbQc are homothetic if and only if P lies on the McCay cubic.
PaPbPc and QaQbQc have the same area if and only if P lies on the McCay cubic.
It is known that the shapes of repeated pedal triangles with respect to a fixed point P recur with period 3. The McCay cubic is the locus of P for which the third pedal is homothetic to the reference triangle.
let P be a point. The parallels at A to PB, PC meet BC at Ab, Ac respectively and Oa is the circumcenter of AAbAc. Define Ob, Oc similarly. The triangles ABC and OaObOc are perspective if and only if P lies on the McCay cubic (together with the line at infinity and the circumcircle) (from Hyacinthos #5618)
Locus of point P such that the pedal triangles of P and its isogonal conjugate P* are orthologic (together with the line at infinity and the circumcircle). See also CL021.
Locus of point P such that the circumcevian triangle of P and ABC are orthologic. The locus of the orthology centers is Q046, the McCay butterfly.
Locus of point P such that the reflections of P in the sidelines of the circumcevian triangle of P form a triangle perspective to ABC (together with the circumcircle).
Locus of point P such that there is an isogonal pK with asymptotes parallel to the cevian lines of P. See CL009. In this case, there is a point M such that the three Simson lines passing through M are parallel to the cevian lines of P. If P = u:v:w this point M is u^2(v+w)/a^2 : : . See more details in Asymptotic Directions of Pivotal Isocubics.
(generalization of 12) Let t be a real number and P, Q two isogonal conjugates. PaPbPc, QaQbQc are the homothetic under h(P,t), h(Q,t) of the pedal triangles of P, Q respectively. PaPbPc and QaQbQc are orthologic if and only if P and Q lie on the McCay cubic (together with the line at infinity and the circumcircle). More generally, when P and Q are two isoconjugates under an isoconjugation with pole W, the locus is a member of the class CL021 of cubics (together with the line at infinity, its W-isoconjugate and the trilinear polar of the isotomic conjugate of the isogonal conjugate of W).
Locus of point P such that the pedal and antipedal triangles of P are orthologic (together with the line at infinity and the circumcircle).
Locus of point P such that the circumcenters of the pedal and antipedal triangles of P are collinear with P or O (together with the line at infinity). With the pedal triangle, the locus of the circumcenter is K258. If we replace circumcenters with centroids, orthocenters, symmedian (Lemoine) points, nine-point centers, we obtain Q017, Q018, Q019, Q020 respectively.
The perpendicular at P to BC meets the line AP* at A'. La is the parallel at A' to AP. Define B', C', Lb, Lc similarly. These lines La, Lb, Lc concur if and only if P lies on the union of the McCay cubic and the orthocubic. Now, if La is the parallel at A' to BC and if Lb, Lc are defined likewise, the lines concur if and only if P lies on the quartic Q023 (Floor van Lamoen and friends, Hyacinthos #5360 & sq.).
Locus of point P such that the reflections of P in the sidelines of ABC form a triangle perspective to the circumcevian triangle of P (together with the circumcircle). The locus of the perspector is Q047, the inversive image of the McCay cubic in the circumcircle.
Locus of point P such that the circumcevian and pedal triangles of P are orthologic (together with the line at infinity and the circumcircle).
Locus of point P such that the circumcevian triangle of P and ABC are orthologic (together with the line at infinity and the circumcircle). The locus of the orthology centers is the McCay cubic itself and Q046, the McCay butterfly. This is connected with bipedal ellipses tritangent to the Steiner deltoid.
Locus of point P such that the antipedal and reflection triangles of P are orthologic (together with the line at infinity and the circumcircle).
Locus of point P such that the circumcevian and reflection triangles of P are orthologic (together with the line at infinity and the circumcircle).
Locus of point P such that the circumcevian triangle of P and the triangle formed by the reflections of P in A, B, C are orthologic (together with the line at infinity and the circumcircle).
Locus of point P such that the circumanticevian and reflection triangles of P are parallelogic.
Let P, P* be two isogonal conjugate points and A'B'C' the circumcevian triangle of P. The locus of P such that the reflections of AP*, BP*, CP* in the respective sidelines B'C', C'A', A'B' of A'B'C' are concurrent is the McCay cubic together with the line at infinity and the circumcircle (Antreas P. Hatzipolakis, Francisco Javier Capitan, Hyacinthos #21687, 21688).
Let P be a point, A*B*C* its circumcevian triangle, A'B'C' its pedal triangle and let A"B"C" be the antipodal triangle of A'B'C' in the pedal circle of P. A*B*C*and A"B"C" are perspective if P is on the circumcircle (the circumcevian triangle degenerates) or on K003 or on K634. If P is on K634, the perspector of the triangles A*B*C* and A"B"C" is a point in the circumcircle (Antreas P. Hatzipolakis, Hyacinthos #21738, Angel Montesdeoca, Hyacinthos #21743).
Locus of the radical center of three circles with same radius, each passing through two vertices of the reference triangle ABC (Lemoine, AFAS 1892, p.102). See "Two groups of points on K003" below.
Locus of point M such that O, M and the isogonal conjugate of M with respect to the circumcevian triangle of M are collinear (together with the circumcircle). See a generalization and related curves at Q050.
Denote by PaPbPc the circumcevian triangle of a point P. The sums of squared distances PaB^2 + PbC^2 + PcA^2 and PbA^2 + PcB^2 + PaC^2 are equal if and only if P lies on K003. See K002, property 26, for an analogous property (Kadir Altintas).
Denote by PaPbPc the circumcevian triangle of a point P and by QaQbQc the circumcevian triangle of the isogonal conjugate of P.The sums of squared distances PaQb^2 + PbQc^2 + PcQa^2 and PbQa^2 + PcQb^2 + PaQc^2 are equal if and only if P lies on K003. (Kadir Altintas).
Denote by PaPbPc the circumcevian triangle of a point P. The antipedal lines of Pa, Pb, Pc concur if and only if P lies on K003. The point of concurrence is the orthocenter of PaPbPc (Angel Montesdeoca, 2021-01-23). Recall that the antipedal line of a point X on the circumcircle is the line passing through X and its isogonal conjugate X*. It is the perpendicular at X to the Simson line of X.

Locus properties related to Simson lines :

Let P, P* be two isogonal conjugate points and A'B'C', A"B"C" the circumcevian triangles of P, P* resp. Denote sA', sB', sC' the Simson lines of A', B', C' wrt A"B"C" resp. These lines sA', sB', sC' are concurrent if and only if P lies on the McCay cubic (together with Kjp, the circumcircle). In other words, if P lies on the McCay cubic then the lines sA', sB', sC' are concurrent (Antreas P. Hatzipolakis, Angel Montesdeoca, Hyacinthos #21685).
Let P, P* be two isogonal conjugate points and A'B'C', A"B"C" the pedal triangles of P, P* resp. (P, P* share the same pedal circle). Denote sA', sB', sC' the Simson lines of A', B', C' wrt A"B"C" resp. These lines sA', sB', sC' are concurrent if and only if P lies on the McCay cubic (together with the circumcircle and the line at infinity (Antreas P. Hatzipolakis, Angel Montesdeoca, Hyacinthos #21686).
See also property 15 above.

Related papers :

McCay Stelloids

Miscellaneous properties

Points on K003

A, B, C where the tangents are the altitudes

I, Ia, Ib, Ic where the tangents pass through O

O (pivot) where the tangent is the Euler line

H (isopivot) where the tangent pass through X(51) and X(1075)

X(1075) = O/H (cevian quotient)

X(13855) = X(1075)* = isogonal conjugate of X(1075)

X(1745) the third point on IH and its extraversions on the lines HIa, HIb, HIc

the isogonal conjugates of these four points

Oa, Ob, Oc (cevians of O) where the tangents pass through X(1075)

N1, N2, N3 vertices of the circumnormal triangle

N1*, N2*, N3* their isogonal conjugates at infinity

imaginary foci of the Brocard ellipse i.e. common points of the Brocard axis and the Kiepert hyperbola

H1, H2, H3 projections of H on the sidelines of N1N2N3. These points lie on the bicevian conic C(G,O) with center X(140) passing through X(125), X(1511), X(2972) and obviously Oa, Ob, Oc.

H1*, H2*, H3* their isogonal conjugates

foci of the inconic with center O, perspector X(69), passing through X(125), X(1565), X(2968). These four points lie on the parallels at O to the asymptotes of the Jerabek hyperbola. See property 2 above.

Seven special points on K003 (Rescassol-Skoubidou points)

These points solve two questions asked independently and almost simultaneously during January 2015 in les-mathématiques.net (in French, thread "Droites parallèles 2015-1") and ADGEOM (thread "Find a point?"). In both cases, the solution was given by Jean-Pierre Ehrmann. Here is a short (slightly rephrazed) summary.

Question 1 : find P such that the pedal circles of P with respect to ABC and the circumcevian triangle of P are identical.

Question 2 : the perpendiculars at P to AP, BP, CP meet the sidelines AB and AC, BC and BA, CA and CB respectively at six points on a same conic. Find P such that this conic is a circle.

Jean-Pierre Ehrmann proved that there are 7 (not always real) such points, interior to the circumcircle and lying on K003. These are not ruler and compass nor conic constructible.

See the mentioned threads for further details.

Another characterization of these points is given in page K634. See also Q157, a circular quintic passing through these points.

Obviously, the isogonal conjugates of these points are seven other points on K003 and on Q158, the isogonal transform of Q157.

In the figure above, there is one and only one real point P. For an almost "flat" triangle ABC, one can find up to five real such points P.

Recall (see property 6) that the pedal triangle PaPbPc of P and the circumcevian triangle A'B'C' of P are homothetic.

Inscribed equilateral triangles

K003 contains nine points on the cevian lines of the vertices of the circumnormal triangle and their nine isogonal conjugates. The former nine points lie on the Apollonius circles and form three equilateral triangles with sidelines perpendicular to those of the circumtangential triangle. These points are three by three collinear with the isodynamic points. A construction is given below.

Compare this configuration and the analogous configuration with the Kjp cubic.

More generally, there are infinitely many equilateral triangles inscribed in the McCay cubic since it is a stelloid with radial center G. Their centers lie on the McCay equilateral quintic Q065.

In particular, the two circles passing through H, X(74) and one of the intersections X(1113), X(1114) of the Euler line and the circumcircle meet the McCay cubic at the vertices of two such triangles.

These circles are orthogonal and centered at O1, O2 on the perpendicular bisector of HX(74) and on the axes of the inconic with center O (these axes are the parallels at O to the asymptotes of the Jerabek hyperbola). O1, O2 also lie on the circle passing through O, H, X(74). Each circle contains two foci of this inconic (these four foci obviously on the McCay cubic).

The two triangles have their sidelines perpendicular and their vertices collinear with H. Since H is the isopivot of the McCay cubic, the vertices of one triangle are the O-Ceva conjugates of the vertices of the other triangle.

Naturally, the isogonal conjugates of these six vertices also lie on the McCay cubic.

K003, the Steiner ellipse and the ITB triangle

K003 meets the Steiner ellipse at A, B, C and three other points M1*, M2*, M3* which are the isogonal conjugates of the common points M1, M2, M3 of the Lemoine axis and K003.

These points M1, M2, M3 are the perspectors of ABC and the ITB triangle A'B'C', the only triply bilogic triangle inscribed in the circumcircle (Jean-Pierre Ehrmann, Hyacinthos #14350).

This means that ABC and A'B'C' are triply perspective and orthologic. M1*, M2*, M3* are three of the six centers of orthology. The three other lie on Q046, the McCay butterfly and also on the related K1098.

K410 and K411 are two pKs passing through A', B', C'.

The following cubics also pass through A', B', C' : K444, K1098, K1100, K1101, K1102, K1103, K1104, K1105 and the isogonal transforms of K056, K1102.

K003 : polar conics, hessian and "negative-hessian"

Recall that the polar conic C(M) of any point M with respect to K003 is a rectangular hyperbola.

The hessian of K003 is K048. It is the locus of point M whose polar conic in K003 degenerates into two perpendicular lines secant at N which also lies on K048. The mapping Psi : M --> N is the product of the reflexion about an axis of the Steiner inellipse and the inversion with respect to the circle centered at G passing through the real foci of this same ellipse. More details in Orthopivotal Cubics and Table 60.

C(M) is a bicevian rectangular hyperbola if and only if M lies on a cubic (K) = K1256 we call the negative-hessian of K003. Indeed, K048 is given by the determinant of the hessian matrix of K003 (see Special Isocubics, §2) and then, when the signs in the diagonal are changed, we obtain another matrix whose determinant is (K).

Properties of (K)

(K) meets the McCay cubic K003 at 9 points namely :

• the in/excenters of ABC, each counted twice since the tangents are the same in both cubics and pass through O. The perspectors of these four bicevian conics are complicated.

• X(1075), the X(3)-Ceva conjugate of X(4), whose polar conic is the bicevian rectangular hyperbola C(X3, X15352) passing through X(4), X(417), X(1075), the vertices A', B', C' of the cevian triangle of X(3).

(K) also contains X(20), X(185), X(14135) and X(14709), X(14710) on the Euler line.

***

The polar conic of X(20) in K003 is the bicevian rectangular hyperbola C(X2, X110) which is at the same time :

• the complement of the Jerabek hyperbola,

• the X(2)-Ceva conjugate of the Brocard axis,

• the polar conic of X(5) in K002.

***

The polar conic of X(185) in K003 passes through X(4), X(281), X(1068) with two complicated perspectors.

***

Note that the polar conics of X(3) in K048 and (K) are the same diagonal rectangular hyperbola (C) passing through the in/excenters, X(661), X(896) and the square roots of every point on the trilinear polar of X(691), passing through {6, 110, 111, 895, 1995, 2493, 2502, 2503, 2854, 2930, 3066, 3124, 5640, 5968, 6593, 7732, 7733, 9129, 9149, 9465, 10545, 10546, 10765, 11188, 11422, 11443, 11447, 11448, 11452, 11453, 11624, 11626, 11746, 12272, 13167, 13410, 15019, 15355, 15531, 16776, 19110, 19111, 19122, 20976, 20998, 21460, 23348, 28662, 32222, 32245, 32246, 32248, 32291, 32292, 32299, 32301, 32302, 32740, 33900, 33928, 35259, 35904, 36696, 37775, 37776, 38851, 39024, 39689, 40228}.

(C) meets the line at infinity at the same points as the rectangular circum-hyperbola with perspector X(2492), center X(5099), passing through X(4), X(23), X(316), X(842), X(1383). This is the isogonal transform of the trilinear polar of X(935), passing through {6, 67, 125, 1560, 5094, 5095, 6103, 6697, 6793, 8791, 15106, 15113, 15116, 15118, 15121, 15128, 15131, 15135, 15140, 15141, 15142, 16176, 17847, 19379, 19504, 26869, 32251, 32252, 32253, 32276, 32280, 32281, 32283, 32285, 32317, 34470, 35325}.

***

(K) belongs to the pencil generated by K048 and the union of the cevian lines of X(3) hence (K) meets K048 at three triads of (not always all real) points.

***

(K) meets the line at infinity at the same points as pK(X6, Z) where Z is the midpoint of X(20) and X(185), a point on the lines {X3,X64}, {X22,X1204}, {X30,X143}, {X51,X3146}, {X52,X1657}, {X72,X2808} and many others, SEARCH = 15.4782303846474. Z is now X(46850) in ETC.

The six remaining finite common points lie on the diagonal rectangular hyperbola (C') passing through the in/excenters, X(371), X(372), X(1707), X(1724), X(1754) and more generally, the square roots of every point on the trilinear polar of X(112), passing through {6, 25, 51, 154, 159, 161, 184, 206, 232, 1194, 1474, 1495, 1660, 1843, 1915, 1971, 1974, 2194, 2203, 2299, 2393, 2445, 3192, 3981, 5320, 5410, 5411, 5412, 5413, 5594, 5595, 6467, 7716, 8541, 8739, 8740, 8779, 8880, 8881, 9418, 9604, 9777, 9924, 9969, 9971, 9973, 10311, 10533, 10534, 10602, 10641, 10642, 10985, 11134, 11137, 11216, 11241, 11242, 11243, 11244, 11388, 11389, 11402, 11405, 11408, 11409, 12167, 12367, 13366, 14153, 14576, 15004, 15010, 15011, 15652, 17809, 17810, 17813, 17819, 17820, 17826, 17827, 18374, 19005, 19006, 19118, 19125, 19132, 19136, 19153, 19355, 19356, 19363, 19364, 19459, 19558, 19596, 20987, 21637, 21639, 21640, 21641, 21647, 21648, 22135, 22829, 23347, 26864, 30402, 30403, 31860, 32366, 32621, 34397, 34417, 34515, 34516, 34565, 34566, 34750, 34751, 34777, 35901, 36879, 37083, 37538, 39125, 39690, 40114}.

(C') also passes through the infinite points of the rectangular circum-hyperbola with perspector X(2485), center X(127), passing through X(4), X(22), X(251), X315), X(1297). This is the isogonal transform of the trilinear polar of X(44766), passing through {3, 66, 141, 157, 159, 160, 1352, 1503, 2353, 4048, 5938, 6247, 6389, 6696, 7789, 7795, 7800, 8721, 9833, 12134, 14216, 14376, 14634, 15270, 15577, 15578, 15579, 15580, 15581, 15582, 15585, 15594, 15812, 18307, 18437, 18440, 18642, 20208, 21167, 23172, 23208, 23328, 32459, 33582, 33926, 34107, 34118, 34138, 34780, 34782, 34828, 34850, 35228, 35707, 36989, 37813, 38909, 39879}.

See page K1256 for further properties.

K003 : prehessians

K003 has three real prehessians P1, P2, P3. In other words, the hessians of P1, P2, P3 are K003. Naturally, K003, its hessian and these three curves share the same nine inflexion points. They belong to a same pencil of cubics called syzygetic pencil of cubics.

These cubics Pi are associated to three new mappings Fi defined as follows. Fi maps any point M to the center of the polar conic of M in Pi.

Fi is an involution of the plane that leaves K003 globally unchanged.

Note that the polar conics of all the points of the Brocard axis in Pi are rectangular hyperbolas. This shows that the prehessians Pi share the same orthic line, namely the Brocard axis.

The polar conics of the centroid G in P1, P2, P3 are the Apollonian circles passing through A, B, C respectively.

The three singular points of F1 are the vertices of the equilateral triangle AaAbAc inscribed in the A-Apollonian circle. See above. This latter circle is actually the image of the Brocard axis under F1. Recall that these points Aa, Ab, Ac, etc, being singular must lie on the McCay cubic.

More generally, F1 transforms any line through O into a conic passing through A and Aa, Ab, Ac and, in particular, the Euler line into a rectangular hyperbola passing through the A-cevian Oa of O, the center Ωa of the A-Apollonius circle. The tangent at A is the altitude AH and the axes are the parallels at the midpoint of AΩa to the bisectors at A of ABC. This gives a construction of Aa, Ab, Ac.

***

The poloconic of a line L in Pi is a conic tritangent to K003 at the images of the common points of L and K003 under Fi.

If L is tangent to K003, the conic is a surosculating conic and is tangent to the McCay cubic at another point. For example :

– if L is the tangent at O to K003 (the Euler line), the conics are surosculating at A, B, C and pass through Oa, Ob, Oc,

– if L is the tangent at I to K003 (the line IO), the conics are surosculating at the excenters Ia, Ib, Ic and pass through A, B, C.

If L is an inflexional tangent at P to K003, the conic is a sextactic conic. The poloconic of L has a sextuple contact with K003 at Fi(P).

***

Feuerbach theorem in K003

There are four lines (not always real) such that their poloconics with respect to one of the three prehessians Ωi are circles.

In such case, these circles are tritangent to K003 and tangent to the corresponding Apollonius circle.

This is the generalized Feuerbach theorem for the McCay cubic : the Apollonius circle is analogous to the nine point circle and the four circles are analogous to the in/excircles.

The figure shows a configuration where the four circles are real and tangent to the B-Apollonius circle.

Parallels to the asymptotes of K003

If an equilateral triangle with center G is drawn with sidelines parallel to the asymptotes of K003 then these sidelines meet K003 on the line at infinity and at six other points lying on a same circle whose center Ω lies on the perpendicular at G to the Brocard axis.

When the sidelines concur at G, this circle splits into the line at infinity and its satellite which is parallel to the Brocard axis. This line is the homothetic of the Brocard axis under h(G, 2/3).

If three parallels to the asymptotes of K003 are drawn through a point P then these parallels meet K003 on the line at infinity and at six other points lying on a same rectangular hyperbola (H1) whose center O1 is the midpoint of PO2 where O2 is the center of the rectangular hyperbola (H2) which is the polar conic of P in K003.

These two rectangular hyperbolas have the same asymptotic directions therefore they are homothetic. Their two other (finite) common points lie on the polar line (L1) of P in (H1).

This line (L1) is also the homothetic of the polar line (L2) of P in (H2) and in K003 under h(P, 2/3).

When the parallels concur at G, this hyperbola splits into the line at infinity and its satellite.

Two groups of points on K003

As in property 29 above, consider six circles with same radius defined as follows. (C1) passes through B, C with center in the half-plane not containing A and (C'1) is its reflection about BC. The circles (C2), (C'2) and (C3), (C'3) are defined likewise.

For i, j, k in {1,2}, define the point ijk as the radical center of three circles as Lemoine did. For example 111 is the radical center of (C1), (C2), (C3) and 112 is the radical center of (C1), (C2), (C'3).

A group of eight (green) points is thus defined and these points all lie on the McCay cubic.

If the common radius of the circles is t R (t > 0, R circumradius), then the trilinear coordinates of ijk are :

cos A – i √(t^2 – sin^2 A) :

cos B – j √(t^2 – sin^2 B) :

cos C – k √(t^2 – sin^2 C).

Collinearities related to sums modulo 3

Two points can be added (digit by digit) modulo 3 to obtain a certain sum. For example 111 + 112 = 220, 111 + 212 = 020, etc.

The results thus obtained can be used to identify several collinearities on K003.

• two points are collinear with O (hence isogonal conjugates) when the sum is 000. Examples : 111 and 222, 112 and 221, etc.

• two points are collinear with A when the sum is 0jk. Examples : 111 and 211 (sum 022), 112 and 212 (sum 021), etc.

• similarly, when the sum is i0k or ij0, we obtain points collinear with B or C.

From this, we see that these eight points lie two by two on four groups of four lines passing through O, A, B, C respectively.

It is clear that each sum of two distinct points must always contains at least one zero hence it remains to see the case when a sum contains two zeros. This gives the second group of six points labelled Yn, Zn on K003 with n in {1, 2, 3}.

For example, Y1 is the intersection of the lines 111-122, 212-221 (sums 200, 100) and Z1 is the intersection of the lines 222-211, 112-121 (sums 100, 200).

The triangles Y1Y2Y3, Z1Z2Z3 are perspective at H, and perspective to ABC at Y, Z on K003. Moreover, Y, Z, X(1075) are collinear on K003.

Note that Y2Z3, Y3Z2 meet at A', the A-cevian of O, and similarly with B', C'.