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X(1), X(2), X(3), X(4), X(6), X(9), X(57), X(223), X(282), X(1073), X(1249), X(3341), X(3342), X(3343), X(3344), X(3349), X(3350), X(3351), X(3352), X(3356)

excenters Ia, Ib, Ic and more generally the extraversions of all the weak points such as X(9), X(57), etc

midpoints of ABC, midpoints of altitudes

E(1986), E(1987) : real foci of the inscribed Steiner ellipse

Q1, Q2, Q3 vertices of the Thomson triangle

other points below

The Thomson cubic is the isogonal pK with pivot the centroid G = X(2). See Table 27. It is sometimes called 17-point cubic in older literature. It is the complement of the Lucas cubic. See Table 21 for cubics anharmonically equivalent to the Thomson cubic. K002 is a member of the class CL043 : it meets the circumcircle at A, B, C and three other points Q1, Q2, Q3 lying on several rectangular hyperbolas (see below). The tangents at these points are concurrent at a point X on the lines X(2)X(1350) and X(64)X(631). See K346 and also Q063.

Locus properties :

1. Denote by PaPbPc the pedal triangle of point P. Under the symmetry in P, PaPbPc is transformed into QaQbQc which is perspective to ABC if and only if P lies on the Thomson cubic. For this reason, the Thomson cubic is called –1-pedal cubic in Pinkernell's paper. The perspector lies on the Lucas cubic which is the 1-cevian cubic.
2. Locus of perspectors (or centers) of circum-conics such that the normals at A, B, C concur (on the Darboux cubic).
3. Let P be the perspector of a circum-conic (C) with center Q. The circum-conic (C') passing through P and Q is a rectangular hyperbola if and only if P lies on the Thomson cubic. In this situation, the axes of (C) are parallel to the asymptotes of (C'). Furthermore, the normals at A, B, C to (C) concur on the Darboux cubic as seen above. See also K172.
4. Locus of point P whose anticevian triangle is orthologic to ABC. The centers of orthology are two points of the Darboux cubic symmetric about O.
5. A' is the second intersection of the line PA with the circle PBC and B', C' are defined similarly. The (perspective) triangles are homothetic if and only if P lies on the Thomson cubic. (Hyacinthos #5879-80-83)
6. 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 orthologic if and only if P lies on the union of Kjp and the Thomson cubic. (from Hyacinthos #5618)
7. Locus of point P such that the trilinear polar and the polar line (in the circumcircle) of P are parallel.
8. Locus of point P such that the polar lines (in the Steiner circum-ellipse) of P and its isogonal conjugate P* are parallel.
9. Locus of point P such that the polar lines (in the Steiner in-ellipse) of P and its isogonal conjugate P* are parallel.
10. Ix is one in/excenter and A'B'C' is the medial triangle. For any point M, the parallel at Ix to BC meets MA' at Ma, Mb and Mc similarly. The triangles ABC and MaMbMc are perspective if and only if M lies on the Thomson cubic. The locus of the perspector is K034 (Philippe Deléham, 16 nov. 2003). Jean-Pierre Erhmann observes that this can easily be generalized when Ix is replaced by any point P and G by any point Q in the following manner : let A'B'C' be the cevian triangle of Q and A", B", C" the traces of the trilinear polar of Q. Let Ma = PA" /\ MA', Mb and Mc similarly. The triangles ABC and MaMbMc are perspective if and only if M lies on pK(P<->P,Q) = pK(P^2,Q) and the locus of the perspector is pK(Q<->Q,P*) where P* is the image of P under the isoconjugation with fixed point Q. For example, with P = X(6) and Q = X(2), the loci are K177 and K141 respectively.
11. A'B'C' is the medial triangle and PaPbPc is the pedal triangle of point P. The locus of P such that PaA'/PaP + PbB'/PbP + PcC'/PcP = 0 (signed distances) is the Thomson cubic. (François Lo Jacomo, communicated by Philippe Deléham, 17 nov. 2003)
12. The parallels at point P to each sideline of ABC meet the other sidelines at six points which always lie on a same conic with center S. The line PS contains the Lemoine point X(6) if and only if P lies on the Thomson cubic. More generally, the line PS contains Q if and only if P lies on pK(Q, X2). For example, when Q = X(187), X(3), X(32) we obtain the cubics K043, K168, K177 respectively.
13. Denote by A1, B1, C1 the reflections of P in A, B, C and by A3, B3, C3 the reflections of H in the lines AP, BP, CP. The three circles AB1C1, BC1A1, CA1B1 have a common point D on the circumcircle. The three circles PAA3, PBB3, PCC3 have P in common and another point L also on the circumcircle. These points D, L are antipodes if and only if P lies on the Thomson cubic (Musselman, Some loci connected with a triangle. Monthly, p.354-361, June-July 1940). On another hand, they coincide if and only if P lies on the bicicular quartic Q013 we call the Musselman quartic.
14. The trilinear polar of P meets the sidelines of ABC at U, V, W. Ub, Uc are the projections of U on AC, AB and Vc, Va, Wa, Wb are defined similarly. These six points lie on a same conic if and only if P lies on the Thomson cubic (together with three circum-conics with perspectors the infinite points of the altitudes corresponding to the case where two points on one sideline coincide).
15. The algebric areas of triangles UbVcWa and UcVaWb are opposite if and only if P lies on the Thomson cubic (Jean-Pierre Ehrmann). They are equal if and only if P lies on K231.
16. These triangles UbVcWa and UcVaWb are parallelogic if and only if P lies on the Thomson cubic or on the cubic K232. They are orthologic if and only if P lies on K232 or on K233 = pK(X25, X4). In the case of P on K232, the points Uc, Va, Wb are in fact collinear.
17. Locus of point P such that the trilinear polar of P is perpendicular to the line PO. See CL040 for a generalization.
18. Locus of the {X}-anticevian points where X is a center on the Lucas cubic. See Table 28 : cevian and anticevian points.

The following table gives the repartition of points on K002 : for any P on K002, the points P*, Q= G/P, R, T are its isogonal conjugate, G-Ceva conjugate, K-crossconjugate, tangential respectively and all these points lie on K002.

Recall that the triples P, P*, X(2) - P, Q, X(6) - P, R, X(3) - Q, G/P*, X(4) are collinear on K002.

Construction of T : let N be the harmonic conjugate of G with respect to P and P* and let N* be its isogonal conjugate. The tangents at P and P* pass through N*. T also lies on the line passing through G/P* and (G/P)*.

The green points are weak points hence their extraversions lie on K002.

E(1986) and E(1987) are the real foci of the Steiner inscribed ellipse. When a point is not in ETC, a SEARCH number is given.

Thomson cubic and Steiner in-ellipse

K002 contains :

• the four foci of the Steiner in-ellipse (S)
• the midpoints of ABC
• the feet R1, R2, R3 of the normals drawn from K to (S) (the fourth foot is X(115), center of the Kiepert hyperbola).

These points R1, R2, R3 lie on the sidelines of Q1Q2Q3, triangle formed by the intersections of K002 and the circumcircle we shall call the Thomson triangle. More informations below.

R1, R2, R3 also lie on :

• (A), the Apollonius hyperbola of K with respect to (S) passing through X(2), X(4), X(6), X(39), X(115), X(1640), X(3413), X(3414).
• The circle with center the midpoint of X(6)X(376) passing through the reflection X(2482) of X(115) in G.

Obviously, the lines GQi and KRi are parallel.

Q1, Q2, Q3 also lie on each rectangular hyperbola of the pencil generated by :

• (H1) passing through X(2), X(3), X(6), X(110), X(154), X(354), X(392), X(1201), X(2574), X(2575),
• (H2) passing through X(2), X(511), X(512), X(574), X(805).

G is the orthocenter of Q1Q2Q3 hence (H1) is the Jerabek hyperbola of Q1Q2Q3.

See a generalization in :

How pivotal cubics intersect the circumcircle

The tangents at these points Q1, Q2, Q3 concur at X on the lines X(2)X(1350) and X(64)X(631). X is the Lemoine point of the Thomson triangle Q1Q2Q3.

The first coordinate of X is :

a^2[a^2(2b^2+2c^2+a^2)-3(b^2+c^2)^2-20b^2c^2].

Its SEARCH number is : 3.2990623265.

The two triangles Q1Q2Q3 and R1R2R3 are perspective at Y on the lines X(6)X(376) and X(39)X(631) with first coordinate :

3a^2(a^2+4b^2+4c^2)+(b^2-c^2)^2

and SEARCH number : 1.3852076515

The tangents at R1, R2, R3 to K002 concur at H.

The Thomson triangle

Recall that we call Thomson triangle the triangle T whose vertices are the three points where the Thomson cubic meets the circumcircle of ABC again.

Here is a list of properties of this triangle, some of them already mentioned above.

1. ABC and T share the same circumcircle and the same Euler line.
2. The Steiner inellipse of ABC is also inscribed in T.
3. The orthocenter of T is G, its incenter is X(5373), its centroid is X(3524), its nine point center is X(549).
4. The Simson line of each vertex of T is parallel to the opposite sideline. The triangle formed with these three Simson lines is the reflection of T about G. It follows that the Steiner inellipse is also inscribed in this triangle.
5. The Jerabek hyperbola of T is the conic passing through X(2), X(3), X(6), X(110), X(154), X(354), X(392), X(1201), X(2574), X(2575).
6. The Kiepert parabola of T has its focus at X(74) and its directrix is the Euler line. It is the reflection about O of the Kiepert parabola of ABC.

More informations here.