Q002


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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 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.

Q002 is the locus of P such that :

the trilinear polar of P* and the orthotransversal of P* are parallel together with the circumcircle (P* is the isogonal conjugate of P).
P lies on the Euler line of its pedal triangle. See also Q039.
PP* is perpendicular to the trilinear polar of P*. (Hyacinthos #2683, #2753)
the orthopivotal cubic O(P) contains P*.
P, P*, H/P* (cevian quotient) are collinear.
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)
P lies on the tangent at P* to the rectangular circum-hyperbola through P*.
the isogonal conjugate of oc(P*) lies on the line KP or, equivalently, oc(P*) lies on the circumconic through G and P*.
O, P, P/P* are collinear.
G, the O-isoconjugate of P, the isotomic conjugate of the isogonal conjugate of the barycentric square P^2 of P are collinear.
K, P^2, the X(184)-isoconjugate of P are collinear.

96 points on Q002

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

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 Z = a^2 SA (b^2+c^2-2a^2) : :
2	X(2574), X(2575) (at infinity)	Q002 has two real asymptote parallel at Z 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 Z
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.

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 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.

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.

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.

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.

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.