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A, B, C which are inflexion points

X(1), X(3), X(6), X(15), X(16), X(358), X(1135), X(1137), X(1155), X(2574), X(2575)

excenters, extraversions of X(1155)

6 feet of bisectors

common points of the Thomson cubic and the circumcircle i.e. vertices of the Thomson triangle

26 mates of X(358) (these are the isogonal conjugates of the perspectors of ABC and the 27 Morley triangles)

See details, figures and other points below

The Euler-Morley quartic Q002 is called Q2 in "Orthocorrespondence and Orthopivotal Cubics" (see Downloads page) where a more complete description can be found. Its isogonal conjugate is the Euler-Morley quintic Q003. See also the analogous quartic Q043 and Q067 and a generalization in CL009.

Locus properties :

Q002 is the locus of P (with isogonal conjugate P*) such that :

  1. the trilinear polar of P* and the orthotransversal of P* are parallel (together with the circumcircle).
  2. P lies on the Euler line of its pedal triangle. See also Q039.
  3. PP* is perpendicular to the trilinear polar of P*. (Hyacinthos #2683, #2753)
  4. the orthopivotal cubic O(P) contains P*.
  5. P, P*, H/P* (cevian quotient) are collinear.
  6. the reflections A", B", C" of A, B, C in the sidelines of the circumcevian triangle A'B'C' of P form a triangle perspective to A'B'C'. (Paul Yiu, Hyacinthos #8631)
  7. P lies on the tangent at P* to the rectangular circum-hyperbola through P*.
  8. the isogonal conjugate of oc(P*) lies on the line KP or, equivalently, oc(P*) lies on the circumconic through G and P*.
  9. O, P, P*/P are collinear.
  10. G, the O-isoconjugate of P, the isotomic conjugate of the isogonal conjugate of the barycentric square P^2 of P are collinear.
  11. K, P^2, the X(184)-isoconjugate of P are collinear.
  12. the line PP* contains the centroid of the pedal triangle of P.
  13. the line PP* is parallel to the Euler line of the pedal triangle of P.

Other properties :

  1. 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 Q002 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.
Q002construction

The trilinear polar of O meets BC at Oa. A variable line (L) passing through Oa meets AB and AC at B' and C'.

C(B) is the conic passing through B with tangent OB, B' and the feet of the bisectors at B on the sideline AC. C(C) is defined likewise.

C(B) and C(C) meet at four points Q1, Q2, Q3, Q4 which lie on Q002.

The pencil of conics generated by C(B) and C(C) contains an analogous conic C(A) passing through A.

It also contains a rectangular hyperbola H(P) passing through O, K, X(2574), X(2575). See below for another point of view related with cubics of the Euler pencil.

See also A Remarkable Rational Transformation Related with Pivotal Cubics.

This construction can be generalized for a certain analogous quartic Q(P, Q) where O = X(3) is replace with P, the feet of the bissectors with the vertices of the cevian triangle of Q and the traces of the trilinear polar of Q.

In this case, if M* denotes the isoconjugate of M under the isoconjugation with fixed point Q (therefore with pole Q^2), then Q(P, Q) is the locus of M such that P, M, M*/M or M, M*, P*/M* are collinear.

With P = u:v:w and Q = p:q:r the equation of Q(P, Q) is : u p^2 (r^2 y^2 - q^2 z^2) y z = 0.

For instance Q002 = Q(X3, X1), Q033 = Q(X3, X2), Q043 = Q(X6, X1), Q045 = Q(X8, X2), Q083 = Q(X4, X1).

96 points on Q002

The following table sums up the points that lie on this remarkable quartic (n is the number of associated points). See an explanation below.

n

description

notes

3

A, B, C

inflexion points with tangents passing through O

2

circular points at infinity

the singular focus of Q002 is X(23), not on the curve

1

O circumcenter

the tangent at O contains X(49) and X(3292) = a^2 SA (b^2+c^2-2a^2) : :

2

X(2574), X(2575) (at infinity)

Q002 has two real asymptotes parallel at X(3292) to those of the Jerabek hyperbola

4

in/excenters

the tangents at these points concur at O

4

X(1155) and extraversions

these points lie on the lines through O and an in/excenter

1

K symmedian point

the tangent at K contains X(373) and X(3292)

6

feet of the bisectors

 

6

other points on the symmedians

these points also lie on the circles centered at the vertices of the tangential triangle passing through the corresponding two vertices of ABC

2

isodynamic points X(15), X(16)

the tangents at these points pass through X(23)

3

circumcircle points

intersections (other than A, B, C) of the Thomson cubic and the circumcircle. The tangents at these points concur at E(227), intersection of the Euler line and the line KX(373).

4

isogonal of CPCC points

these points on K004 and K172, see Table 11.

4

isogonal of Ix-anticevian points

these points on K005 and K073, see Table 23.

4

foci of the ellipse K

this is the inconic with center K

27

isogonal of Morley perspectors

among them X(358), X(1135), X(1137). See Table 9.

3

other points on the Euler line

these three points lie on K019

3

other points on the circum-conic through X(15), X(16)

these three points lie on K316

9

cube roots of X(184)

this is a consequence of property 11

8

points on the lines through K and an in/excenter

these points lie on the circum-conics with perspectors a(b-c)(b+c-2a)SA : : and extraversions. This point is the barycentric product X69 x X1635 or X63 x X900.

Q002 and the isogonal conjugates of the Morley perspectors

Q002 contains the isogonal conjugates of the 27 perspectors of ABC and the Morley triangles. See Table 9.

These are the points with barycentric coordinates :

a cos (A/3 + k1 2pi/3) : b cos (B/3 + k2 2pi/3) : c cos (C/3 + k3 2pi/3) where k1, k2, k3 are integers in {-1;0;1}. These 27 points lie on three groups of 9 lines passing through A, B, C.

In particular, Q002 contains :

X(358) obtained with k1=k2=k3=0, X(1135) obtained with k1=k2=k3=1, X(1137) obtained with k1=k2=k3=-1. Note that the points X(16), X(358), X(1135), X(1137) are collinear.

Q002X358
Q002X358b
Q002K006

Q002 and the polar curves

of the circumcenter O

The polar line of O is the tangent at O passing through X(49) and the intersection Z = X(3292) of the real asymptotes.

The polar conic of O is the Jerabek hyperbola.

The polar cubic of O is the Orthocubic. The tangents at A, B, C, O are common to both curves.

Q002K004

Q002 and the isogonal conjugates

of the CPCC points

These points are described in Table 11. They are the dark green points on the figure.

They lie on the Darboux cubic, K172 = pK(X32, X3) and several other curves.

Note that K172 meets the circumcircle at the same points as Q002 and the Thomson cubic. These are the vertices of the Thomson triangle. The tangents to Q002 at these points and at X(6) concur at E(227) = X(2)X(3) /\ X(6)X(373).

Q002K005a

Q002 and the isogonal conjugates

of the Ix-anticevian points

These points are described in Table 23. They are the blue points on the figure.

They lie on the Napoleon cubic, K073 = pK(X50, X3) and several other curves.

Recall that K073 is a circular cubic passing through the isodynamic points.

Q002K040

Q002 and the ellipse K

Q002 contains the foci of the ellipse K i.e. the in-conic (K) with center K. It is an ellipse when the triangle ABC is acute angle. They are the red points on the figure.

These four foci also lie on the Pelletier strophoid K040 and many other curves.

Note that Q002 and K040 meet at 12 known points since both curves are circular and contain X(1155) on the line IO.

The vertices of the yellow triangle are the extraversions of X(1155) which lie on Q002.

Q002 and the cubics of the Euler pencil

The Euler pencil is formed by the isogonal pivotal cubics with pivot P on the Euler line. See Table 27.

The frequent occurence of these cubics in the table above can be explained (and extended) by the following decomposition of Q002.

Let K(P) be the cubic pK(X6, P) and H(P) be the rectangular hyperbola passing through O, K, P, X(2574), X(2575) hence having its asymptotes parallel to those of the Jerabek and Stammler hyperbolas which are two members of the pencil.

K(P) and H(P) meet at six points namely O, P and four other points on Q002. Indeed, an easy computation shows that :

Q002 = (a^2 + b^2 + c^2)(x + y + z) K(P) + (a^2 y z + b^2 z x + c^2 x y) H(P).

Recall that Q002 and K(P) have 8 fixed common points A, B, C, X(1), X(3), the excenters, and then 4 more depending of P which must lie on H(P).

Special cases :

• P = O : H(P) is the Stammler hyperbola (tangent at O to the Euler line) and K(P) is the McCay cubic K003. The four points are the in/excenters and the tangents at these points are common to Q002 and K003, passing through O.

• P = H : H(P) is the Jerabek hyperbola and K(P) is the Orthocubic K006. The four points are A, B, C, O and the tangents at these points are common to Q002 and K006, also passing through O.

• P = X(20) : K(P) is the Darboux cubic K004 and the four points are the isogonal conjugates of the CPCC points.

• P = X(5) : K(P) is the Napoleon cubic K005 and the four points are the isogonal conjugates of the Ix-anticevian points.

• P = G : K(P) is the Thomson cubic K002 and the four points are X(6) and the vertices of the Thomson triangle.

• P = X(30) : K(P) is the Neuberg cubic K001 and the four points are the isodynamic points X(15), X(16) and the circular points at infinity. This property is generalized below.

Q002 and the circular pivotal cubics

Q002droiteK

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

Since it already has 9 common points with Q002 (namely A, B, C, in/excenters, circular points at infinity), it must meet Q002 again at three other points Q1, Q2, Q3 which lie on a same line (L) passing through the Lemoine point K.

This line (L) is actually the trilinear polar of the antipode Q of P* on (O).

For example, with P = X(30), the cubic is K001 and (L) is the Brocard axis. The three points are X(3), X(15), X(16) as said above.

Conversely, a line (L) passing through K has its trilinear pole Q on (O) and meets Q002 again at three points which lie on the isogonal circular pK whose pivot P is the isogonal conjugate of the antipode of Q.