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∑ a^2(SB y - SC z) y^2 z^2 = 0

∑ SA x^3(c^2 y^2 - b^2 z^2) = 0

A, B, C which are singular,

X(1), X(2), X(4), X(13), X(14), X(357), X(1113), X(1114), X(1134), X(1136), X(1156)

excenters, feet of altitudes, extraversions of X(1156), infinite points of the Thomson cubic

26 mates of X(357) (these are the perspectors of ABC and the 27 Morley triangles) among them X(1134), X(1136). See Table 9.

CPCC or H-cevian points, see Table 11

Ix-anticevian points, see Table 23

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

intersections of a median and the circle with diameter the corresponding side of ABC

More points, details, figures and generalization below.

***

The Euler-Morley quintic Q003 is called Q1 in "Orthocorrespondence and Orthopivotal Cubics" (see Downloads page) where a more complete description can be found. Its isogonal conjugate is the Euler-Morley quartic Q002 therefore its construction derives easily from that of Q002 given there.

Let P be a variable point, H(P) the rectangular circum-hyperbola passing through P, T(P) its tangent at P.

K005 is the locus of points whose polar conic in Q003 is a rectangular hyperbola. In other words, the cubic is the Laplacian of the quintic.

Locus properties :

Q003 is the locus of P such that :

1. the trilinear polar of P and the orthotransversal of P are parallel.
2. P lies on the Euler line of the pedal triangle of P* = isogonal conjugate of P.
3. the line PP* is perpendicular to the trilinear polar of P or to the orthotransversal of P.
4. the orthopivotal cubic O(P*) contains P.
5. L(P) also contains P and Pcev (isogonal of P in the cevian triangle of P). Recall that for any P in the plane, the points P*, H/P (Ceva conjugate), oc(P) (orthocorrespondent of P) and Pacv (isogonal of P in the anticevian triangle of P) are collinear on a line L(P). See the analogous quintic Q042 and also Table 23.
6. the inconic with focus P has its directrices parallel to the trilinear polar of P.
7. P* or oc(P) lies on T(P). In this case, T(P) is the Euler line of the pedal triangle of P*. See property 2.
8. oc(P) lies on the circum-conic through G and P or, equivalently, the points K, P*, oc(P)* are collinear.
9. O, P* and P/P* are collinear.
10. P lies on the polar line of P* in H(P).
11. the lines GP and T(P) meet on H(P).
12. G, P, Q are collinear, where Q is the second point of H(P) which is on the parallel at H to T(P). Q is the P-isoconjugate of oc(P).
13. H, P, P' are collinear, where P' is the second center of orthology of the anticevian and antipedal triangles of P.
14. G, P and the orthocenter of the antipedal triangle of P are collinear.
15. P' lies on the Euler line of the antipedal triangle of P.
16. the line PP* is parallel to the Euler line of the pedal triangle of P*.
17. H, P and the X(25)-isoconjugate of P^2 are collinear.
18. K, P^2 and the P^3-isoconjugate of H are collinear.
19. the line PP* contains the centroid of the pedal triangle of P*.

Other properties :

1. Let Q be a fixed point. The locus of P such that the line QP* (P* = isogonal of P) is perpendicular to the trilinear polar of P is a quartic C(Q). See Q066 obtained with Q = O. This quartic C(Q) contains Q if and only if Q lies on Q003. (Angel Montesdeoca, private message 2013-11-30).
2. Let ABC be a triangle and P a point. The line AP intersects the circumcircle of the triangle PBC at P and A'. Similarly define B', C'. The Euler lines of triangles BCA', CAB', ABC' are concurrent if and only if P lies on Q003 (together with the circumcircle). See K001 = Neuberg cubic (locus property 12) and Q030 for other loci related to the same configuration. See also the related quartic Q056, Table 28 : cevian and anticevian points, Table 29 : Q-Ix-(anti)cevian points.
3. Let P be a fixed point The Euler line of the pedal triangle of M contains P if and only if M lies on a circular circum-quartic Q(P) passing through X(15), X(16), X(2574), X(2575). This quartic contains P* when P lies on Q003 or on the line at infinity. In the latter case, it decomposes into the line at infinity and a circular cubic which is the isogonal transform of the orthopivotal cubic O(P) whose singular focus is the centroid G of ABC.
4. Q003 is the locus of the common points of the cubics C(k), D(k) when k varies. See On two Remarkable Pencils of Cubics of the Triangle Plane. See also a generalization at the bottom of this page.
5. Let M be a point on the orthic axis. H(M) is the rectangular hyperbola with perspector M and K(M) is the isogonal focal cubic nK0(X6, M), see CL025. When M traverses the orthic axis, H(M) and K(M) meet at A, B, C and three other points lying on Q003. For example, with M = X(523), H(M) is the Kiepert hyperbola and K(M) is K018. The three points are X(2), X(13), X(14). Note that K(M) contains the foci of the K-ellipse.
6. Let P be a point and Na, Nb, Nc the nine-point centers of PBC, PCA, PAB respectively. If P lies on Q003 then the Euler lines of NaNbNc and ABC concur at the nine-point center X(5) of ABC (Angel Montesdeoca, 2017-06-29). This property is also true when nine-point centers and X(5) are replaced with circumcenters and X(3).
7. Let P be a point with cevian triangle PaPbPc. Denote by A1, B1, C1 the inverses of Pa, Pb, Pc in the circles PBC, PCA, PAB respectively. The triangles A1B1C1 is perspective to both triangles ABC and PaPbPc when P lies on Q003. See "other property 4" in Q002 for a similar configuration.
8. Let P be a point with cevian triangle PaPbPc. Denote by A1, A2 the points of BC on the circle with center Pa passing through P. Define B1, B2 and C1, C2 similarly. These six points lie on a same conic if and only if P lies on Q003, togther with the circumcircle, the line at infinity, the sidelines of the antimedial triangle (this answers a question raised by Jean-Louis Breuil, 2023-07-23).

Q003 and the union of the line at infinity, the circumcircle, the Kiepert hyperbola generate a pencil of circular trinodal circum-quintics which contains Q050, Q135 and several other curves. See Q050 for further details.

***

Miscellaneous properties of Q003

Q003 and the Morley perspectors

Q003 contains the 27 perspectors of ABC and the Morley triangles. See Table 9.

These are the points with barycentric coordinates :

a sec (A/3 + k1 2pi/3) : b sec (B/3 + k2 2pi/3) : c sec (C/3 + k3 2pi/3)

where k1, k2, k3 are integers in {-1;0;1}.

In particular, Q003 contains :

X(357) obtained with k1=k2=k3=0,

X(1134) obtained with k1=k2=k3=1,

X(1136) obtained with k1=k2=k3=-1.

These 27 points lie on three groups of 9 lines passing through A, B, C.

Q003 and the CPCC points

Q003 contains the CPCC points as seen in Table 11.

These points are common points of the Darboux cubic K004 and the Lucas cubic K007.

They also lie on several other cubics and on the rectangular hyperbola (H) passing through X(5), X(6), X(20), X(69), X(1498), X(2574), X(2575).

See below, cubics of the Euler pencil.

Q003 and the Ix-anticevian points

Q003 contains the Ix-anticevian points as seen in Table 23.

These points are common points of the Napoleon cubic K005, the McCay orthic cubic K049 and the Kn cubic K060.

They also lie on several other cubics and on the rectangular hyperbola (H) passing through X(5), X(6), X(52), X(195), X(265), X(382), X(2574), X(2575).

See below, cubics of the Euler pencil.

Lines and conics (1)

Q003 contains the four foci of the "ellipse K", the inconic with center K. This conic is an ellipse when ABC is acutangle. It is tangent to the sidelines of ABC at the feet of the altitudes, these points on Q003. These four points also lie on Q002.

Q003 contains X(1156) and its extraversions.

X(1156) is the fourth common point of the Feuerbach hyperbola and the circumconic with perspector I (incenter). Note that this latter conic is inscribed in the excentral triangle IaIbIc hence it is tritangent at A, B, C to Q003.

When I is replaced by the excenters, the Feuerbach hyperbola is replaced by the three Boutin hyperbolas and the three circumconics with perspector Ia, Ib, Ic are also tritangent at A, B, C to Q003.

Lines and conics (2)

Q003 meets the Fermat line at the Fermat points X(13), X(14) and three other points. One is always real, labelled F₁ on the figure, and the three points lie on K316 = pK(X6, X110).

Q003 meets the Jerabek hyperbola at A, B, C (each one counted twice), H and three other points. One is always real, labelled J₁ on the figure, and the three points lie on K019, the (third) Brocard cubic. This cubic also contains the foci of the ellipse K.

It follows that Q002 meets the Euler line on this same cubic.

Lines and conics (3)

The circum-conic (C) passing through the incenter I and the centroid G must meet Q003 at two other points. These points are the intersections of the conic and the line (L) passing through X(19) (Clawson point) and X(2093), the reflection of I in X(57).

The three extraversions of (C) and (L) give six more points on Q003. These are the eight blue points on the figure.

Lines and conics (4)

The circum-conic (C') passing through G and its orthocorrespondent X(1992) is tangent at G to Q003. It must have two other common points E1, E2 with Q003. These points lie on :

– the line X(25)X(2930),

– the pivotal isogonal circular cubic pK(X6, X524),

– the orthopivotal central cubic K065,

– the pivotal cubic K209 = pK(X468, X4).

See a generalization below for any pivotal isogonal circular cubic.

Pretangentials of H

The locus of points M such that the polar line of M in Q003 passes through H is the quartic (Q) meeting Q003 at 20 points : A, B, C (counting for 6), H(twice), Ha, Hb, Hc and 9 other points which are the cube roots of X(25). This is a consequence of property 18 above. Recall that X(25) lies on the Euler line and is the pole of the orthic axis in the circumcircle.

Only one of these 9 points is real and is labelled E on the figure. The 8 remaining points are always imaginary. Six of them lie on the lines passing through E and a vertex of ABC and the two other lie on the trilinear polar of E.

These 9 points also lie on the cubic (K) which is the cube root of the Euler line. This has the very simple equation :

(b^2-c^2)SA x^3 + cyclic = 0.

(K) contains G (with tangent the Euler line), X(30), the infinite points of the Steiner ellipse. It has one real asymptote parallel at X(340) to the Euler line and two imaginary asymptotes concurring at X(648).

X2-OAP points

Q003 contains the four X2-OAP points P1, P2, P3, P4.

They are the points whose antipedal triangle has orthocenter X2 = G.

These points also lie

• on the cubics K243, K315, K808,

• on the quintic Q095,

• on the rectangular hyperbola (H) passing through X(2), X(6), X(376), X(2574), X(2575) hence homothetic to the Jerabek hyperbola.

See also Table 53

99 points on Q003

The following table sums up the points that lie on this remarkable quintic (n is the number of associated points).

Q003 and the cubics of the Euler pencil

(See the bottom of Table 23 for a more general discussion)

Any cubic pK1 of the Euler pencil is an isogonal pivotal cubic with pivot P on the Euler line such that OP = t OH (vectors). For any M on pK1, the points P, M, M* are collinear.

Let pK2 be the pK with pivot H, isopivot P hence pole H x P. For any M on pK2, the points P, M, H/M (Ceva conjugate) are collinear.

If M is a common point of pK1, pK2 then P, M, M*, H/M are collinear and, by property 5 above, M must lie on Q003.

The cubics pK1, pK2 contain A, B, C, H, P and four other points Qi on Q003. Hence, each cubic of the Euler pencil is associated with a tetrad of points Qi on Q003.

pK1, pK2 generate a pencil of cubics passing through the points Qi which contains the (not necessarily all distinct) following cubics :

• a third pK (denoted pK3) with isopivot H, pivot on the Jerabek hyperbola, pole on the circum-conic passing through X(2) and X(6).

• a McCay stelloid K003(P) = spK(X3, Q) with Q on the Brocard axis, radial center on the line X(2), X(51). See CL055.

• a circular cubic K001(P) with focus on the line X(2), X(98) which is in fact the orthopivotal cubic O(P). This cubic contains X(13), X(14), X(30) hence the infinite points of the Neuberg cubic K001.

• a cubic K002(P) passing through X(2) and the infinite points of the Thomson cubic K002. The tangent at X(2) is the line X(2)X(6) for both cubics (except when t = 1/3).

• a cubic K004(P) with three concurring asymptotes (on the Euler line at X such that OX = t(t + 1) / (3t - 1) OH) and passing through the infinite points of the Darboux cubic K004.

• a cubic K005(P) passing through X(17), X(18) and the infinite points of the Napoleon cubic K005.

• a cubic K006(P) passing through X(485), X(486) and the infinite points of the Orthocubic K006.

The four points Qi also lie on a rectangular hyperbola H(P) passing through P, X(6), X(2574), X(2575) hence homothetic to the Jerabek and Stammler hyperbolas.

H(P) meets the Euler line again at P' such that OP' = 2t / (3t-1) OH i.e. the point such that the anharmonic ratio (O, H, P, P') = – 1/2.

Example : t = 1/2 hence P = X(5) and P' = X(382).

pk1 = K005 = K005(P), pk2 = K049 = K003(P), pk3 = K060 = K001(P), K004(P) = K127, K006(P) = K122. The points Qi are the Ix-anticevian points.

Q003 and the circular pivotal cubics

Any isogonal circular pK must have its pivot P at infinity and its isopivot P* on the circumcircle (O).

Since it already has 12 common points with Q003 (namely A, B, C counting for 6, in/excenters, circular points at infinity), it must meet Q003 again at three other points R1, R2, R3 which lie on :

• pK2 = pK(H x P, H),

• the orthopivotal cubic O(P),

• the circle (C) with diameter HM where M is the trilinear pole of the Simson line of P*, hence M lies on the Simson cubic K010 and the center of (C) lies on its image (K) under the homothety h(H, 1/2),

• the circum-conic (H) passing through G and M whose perspector is the infinite point of the direction perpendicular to P.

Example 1 : with P = X(524), (H) passes through X(2), X(1992), X(2408), (C) passes through X(2), X(4), X(895), X(2408) hence M = X(2408), pK2 = K209, O(P) = K065. One of the three points is X(2) and the two others are those on the line X(25)X(2930) mentioned in the table above.

Example 2 : with P = X(30), (H) is the Kiepert hyperbola, (C) passes through X(4), X(13), X(14), X(2132), X(2394) hence M = X(2394). The three cubics as above are K001, K059, K952 and the three points are X(4), X(13), X(14).

Generalization of Q003

Let L be a line and M a variable point on L. Let P = p : q : r and Q = u : v : w be two fixed points.

Consider the two pivotal cubics pK1 = pK(P, M) and pK2 = pK(Q x M, Q). These cubics intersect at A, B, C, M and five other points which lie on a quintic which does not depend on L. This is the locus of a variable point S such that S, P ÷ S (barycentric quotient) and Q / S (cevian quotient) are collinear.

This quintic is given by : ∑ p u (w y - v z) y^2 z^2 = 0 or ∑ v w x^3 (r y^2 - q z^2) = 0.

With P = K and Q = H (resp. G) we obtain Q003 (resp. Q042).

More generally, any other point Q on the Kiepert hyperbola gives a quintic also passing through X(1), X(2), X(13), X(14).

A judicious choice of the line L or the point M explains the fact that such or such points lie on Q003.

For example, when M = K, pK1 is the Grebe cubic K102 and pK2 is K233 both containing the four foci of the K-ellipse.