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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), X(14481)

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

foci of the inscribed Steiner ellipse

Q1, Q2, Q3 vertices of the Thomson triangle, see below

points R0, R1, R2, R3 described in the page K755

other points below

Geometric properties :

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.

The isotomic transform of K002 is K184. The Psi transform of K002 is Q119. The symbolic substitution SS{a -> √a} transforms K002 into K363.

The Thomson isogonal transform of K002 is the central cubic K758.

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 the Lemoine point X(5646) of the Thomson triangle, a point lying 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). Also, locus of the centers of inconics such that the normals at the contacts with the sidelines of ABC are concurrent (on the Darboux cubic) and then, the locus of perspectors of these inconics is the Lucas cubic. See Thomson, NAM 1865, p.144, Darboux, NAM 1866, p.95 and Neuberg, Schoute, AFAS 1891, p.113.
3. Let P be the perspector of a circum-conic (C) with center Q, the G-Ceva conjugate of P. 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 and the related property 25.
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.
19. Let Oa, Ob, Oc be the circumcenters of triangles PBC, PCA, PAB. The centroid of OaObOc lies on the line OP if and only if P lies on the Thomson cubic or on the circumcircle of ABC (Angel Montesdeoca, Anopolis #958).
20. Let A'B'C' be the circumcevian triangle of P. The lines A'B' and A'C' meet BC at Ab and Ac. The points Bc, Ba and Ca, Cb are defined likewise and these six points lie on a same conic with center Q. The points X(6), P, Q are collinear if and only if P lies on the Thomson cubic (Angel Montesdeoca, ADGEOM #905, slightly rephrased). Similarly, the points X(3), P, Q are collinear if and only if P lies on the quartic Q098.
21. Locus of perspectors of circum-conics such that the net of circum-cubics passing through the four foci contains a psK. See K007 and K709 for example which are in fact the only pKs of this kind.
22. Two conics, one inscribed and one circumscribed, with the same center P have parallel axes if and only if P lies on the Thomson cubic or on the line at infinity. In this latter case, the conics are parabolas with same point at infinity. See K015, property 4. If "axes" is replaced with "asymptotes", we obtain K219 and if "center" is replaced with "perspector", we find the Stothers quintic Q012.
23. For any P on K002 (but not on a sideline of ABC nor on the Steiner in-ellipse), one can find a circum-cubic passing through the center, the foci and the infinite points of the circum-conic with perspector P. With P = X2, we obtain K715 and with P = X1, we obtain K716.
24. Two homothetic rectangular hyperbolas (i.e. with same infinite points) one circumscribed, the other diagonal in ABC meet at two finite points which lie on K002 and on a line passing through X(6). These points are therefore X(2)-Ceva conjugates.
25. The conic passing through P, the vertices of the anticevian triangle of P and the G-Ceva conjugate of P is a rectangular hyperbola if and only if P lies on the Thomson cubic. See the related property 3.
26. Denote by PaPbPc the circumcevian triangle of a point P. The sums of inverse squared distances 1 / PaB^2 + 1 / PbC^2 +1 / PcA^2 and 1 / PbA^2 + 1 / PcB^2 + 1 / PaC^2 are equal if and only if P lies on K002. See K003, property 31, for an analogous property (Kadir Altintas).
27. Given a triangle ABC, the parallel through a point P to the sideline BC cuts the perpendiculars to BC though B and C at Ba and Ca respectively. Points Cb, Ab, Ac, Bc are defined cyclically, Let A' be the midpoint of Ab,Ac and B', C' likewise. The lines AA', BB', CC' are concurrent (at a point Q) if and only if P lies on the Thomson cubic K002. The point of concurrence lies on the Darboux cubic K004 (Angel Montesdeoca, 2020-10-25, further details here in Spanish).
28. Let P and Q be points isogonal conjugates with respect to the  triangle ABC. Let A1B1C1 be the circumcevian triangle of P and let A2B2C2 be the cevian triangle of Q. The circles (AA1A2), (BB1B2), (CC1C2) are coaxal if and only if P lies on the Thomson cubic or on the circumcircle (Angel Montesdeoca, 2022-01-25).

Related papers

On the Thomson Triangle

Points on the Thomson cubic

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.

When a point is not in ETC, a SEARCH number is given.

P1, P2, P3, P4, P5, P6 are the tangentials of X(3342), X(3343), X(3344), X(3351), X(3350), X(3352) respectively. These points are rather complicated.

Group law on the Thomson cubic

As in the page K004, a group law is defined on K002 with neutral element X(6). For P, Q on K002, P + Q is the X(2)-Ceva conjugate of the third point of K002 on the line PQ.

Each center P on K002 is associated with an integer n. Three points are collinear if and only if the sum of the corresponding integers is 4. T is the tangential of P.

If P and P' are two points with integers n and n' then :

• P and P' are X(6)-cross conjugates hence collinear with X(3) if and only if n + n' = 0,

• P and P' are isogonal conjugates hence collinear with X(2) if and only if n + n' = 2,

• P and P' are X(2)-Ceva conjugates hence collinear with X(6) if and only if n + n' = 4,

When n + n' = 6, the points are collinear with X(4) and when n + n' = 8, the points are collinear with X(1073). This is easily adapted for other sums n + n'.

The tangential T of P corresponds to 4 - 2n hence the X(2)-Ceva conjugate of T corresponds to 2n.

For example, the point P(-9) = X(46985) is the X(6)-crossconjugate of X(3351) and the isogonal conjugate of P(11) = X(46984).

Note that P is a weak point when n is an odd integer.

K002 contains :

• the four foci of the Steiner in-ellipse (S), see Table 48
• 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, now X(5646) in ETC.

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.

Y is now X(14482) in ETC.

Thomson cubic and Steiner in-ellipse

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

The Thomson triangle and the Thomson-Jerabek hyperbola

Recall that we call Thomson triangle the triangle T whose vertices Q1, Q2, Q3 are the three (always real) points where the Thomson cubic meets the circumcircle of ABC again. This triangle is always acutangle, see a proof in §1.6 here. See Table 56 for related curves.

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), X(5544), X(5638), X(5639), X(5643), X(5644), X(5645), X(5646), X(5648), X(5652), X(5653), X(5654), X(5655), X(5656), X(5888), X(6030), X(7712), X(9716). See the related Table 27. Note that this hyperbola is the polar conic of X(376) in K003, also the polar conic of X(3) in K832.
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.
7. The three cubics nK0(X6, Qi) are the only isogonal nK0s whose orthic line coincide with the trilinear polar of the root. Furthermore, this line is also the orthic line of the Hessian of nK0(X6, Qi). It follows that the polar conics of its traces on the sidelines of ABC with respect to a cubic and its Hessian are rectangular hyperbolas intersecting on nK0(X6, Qi).

More informations here.

K346 is the locus of the poles of pKs meeting the circumcircle at A, B, C, Q1, Q2, Q3. This is the case of K167 = pK(X184, X6) and K172 = pK(X32, X3). The locus of the pivots of such cubics is K002 itself and the locus of the isopivots is K172.

***

Three simple hyperbolas

(HA), (HB), (HC) are hyperbolas passing through the vertices of the Thomson triangle.

(HA) is the Thomson isogonal transform of the sideline BC.

(HB), (HC) are defined likewise.

(HA) passes through the reflection A' of A about O with tangent passing through Ka, the A-vertex of the circumcevian triangle of K.

If AA' is trisected by points A" and Oa, then A" lies on (HA) and Oa is the center of (HA).

The asymptotes of (HA) are perpendicular to the sidelines AB, AC.

The axes of (HA) are parallel at Oa to the bisectors at A of ABC.

Two points on BC, harmonically conjugated in B and C, have their Thomson isogonal conjugates symmetric in Oa.

Two points on BC, symmetric about the midpoint of BC, have their Thomson isogonal conjugates on a line parallel to A'Ka.

Let P be a point with cevian triangle PaPbPc. The trilinear polar of the isogonal conjugate of P meets BC at Qa. With P = p : q : r, we have Pa = 0 : q : r and Qa = 0 : b^2 r : - c^2 q. Note that the barycentric product Pa × Qa is the center of the A-Apollonius circle.

Pa and Qa have their Thomson isogonal conjugates on a perpendicular to BC.

A2 and A1 are obtained when P = X(1).

α' and A' are obtained when P = X(6).

A" and α" are obtained when P = X(2).

The following list shows several centers of the Thomson Triangle and their counterparts in ABC (Chris van Tienhoven, Peter Moses).

For example, {1,5373} means that X(1) in Thomson Triangle is X(5373) in ABC.

Those in red remain unchanged, those in blue are swapped.

{1,5373}, {2,3524}, {3,3}, {4,2}, {5,549}, {6,5646}, {20,376}, {24,1995}, {30,30}, {54,5888}, {64,6}, {74,110}, {107,98}, {110,74}, {122,2482}, {125,5642}, {133,6055}, {185,5650}, {186,23}, {265,5655}, {381,5054}, {382,381}, {399,10620}, {403,7426}, {459,5304}, {476,477}, {477,476}, {520,512}, {523,523}, {526,526}, {546,140}, {550,8703}, {924,8675}, {1073,5024}, {1075,6194}, {1113,1113}, {1114,1114}, {1147,4550}, {1204,5651}, {1294,99}, {1300,1302}, {1301,111}, {1304,842}, {1498,1350}, {1657,3534}, {2071,7464}, {2574,2575}, {2575,2574}, {2693,691}, {2777,542}, {2972,9155}, {3090,3523}, {3091,631}, {3146,4}, {3183,9740}, {3346,9741}, {3357,182}, {3426,5544}, {3515,25}, {3516,7484}, {3517,5020}, {3520,7496}, {3529,20}, {3532,154}, {3543,3545}, {3627,5}, {3628,3530}, {3830,5055}, {3853,547}, {4240,7422}, {5073,3830}, {5076,1656}, {5663,5663}, {5895,599}, {5896,112}, {5897,1296}, {6000,511}, {6080,2698}, {6086,804}, {6241,7998}, {6247,597}, {6526,7735}, {6622,4232}, {6759,3098}, {7687,5972}, {7689,8717}, {8057,1499}, {8798,39}, {9033,690}

***

If M is a point with isogonal conjugate M* with respect to ABC then the isogonal conjugate of M with respect to T is M_T*, the centroid of the antipedal triangle of M* in ABC (Randy Hutson).

The following list gives pairs of such points {M, M_T*}, (Peter Moses, updated 2023-07-28).

For example, {1,165} means that the isogonal conjugate of X(1) with respect to the Thomson Triangle is X(165) in ABC. More informations here.

{1,165}, {2,3}, {4,154}, {5,6030}, {6,376}, {9,3576}, {13,34317}, {14,34318}, {15,5463}, {16,5464}, {20,3167}, {22,5654}, {23,5655}, {25,5656}, {30,110}, {35,5659}, {36,5660}, {40,3158}, {54,35885}, {55,5657}, {56,5658}, {57,52026}, {61,49901}, {62,49902}, {64,42452}, {74,523}, {84,34499}, {98,512}, {99,511}, {100,517}, {101,516}, {102,522}, {103,514}, {104,513}, {105,3309}, {106,3667}, {107,6000}, {108,6001}, {109,515}, {111,1499}, {112,1503}, {186,15131}, {187,6054}, {198,5603}, {223,52027}, {262,47052}, {354,3651}, {371,13712}, {372,13835}, {381,7712}, {392,4220}, {476,5663}, {477,526}, {518,1292}, {519,1293}, {520,1294}, {521,1295}, {524,1296}, {525,1297}, {527,28291}, {528,2742}, {529,39635}, {530,9202}, {531,9203}, {532,39636}, {533,39637}, {535,39638}, {536,28474}, {537,28520}, {538,39639}, {539,20185}, {541,9060}, {542,691}, {543,2709}, {544,39640}, {545,28293}, {549,5888}, {550,55038}, {551,37508}, {573,47040}, {574,22712}, {674,44876}, {688,53889}, {689,55005}, {690,842}, {698,30254}, {699,30217}, {726,28469}, {727,28470}, {729,32472}, {736,39629}, {739,28475}, {740,6010}, {741,6002}, {752,28467}, {753,28468}, {754,53885}, {755,32473}, {758,6011}, {759,6003}, {789,55004}, {804,2698}, {805,2782}, {812,12032}, {813,28850}, {814,29009}, {815,29010}, {824,28844}, {825,28845}, {826,29011}, {827,29012}, {830,53892}, {834,45136}, {840,2826}, {841,9003}, {843,2793}, {891,29348}, {898,29349}, {900,953}, {901,952}, {902,24808}, {907,29181}, {912,13397}, {915,15313}, {916,1305}, {917,8676}, {918,28838}, {924,1300}, {925,13754}, {926,2724}, {927,2808}, {928,2723}, {929,2807}, {930,1154}, {932,15310}, {933,18400}, {934,971}, {935,2781}, {972,3900}, {991,47039}, {1113,2575}, {1114,2574}, {1137,11228}, {1141,1510}, {1249,10606}, {1289,34146}, {1290,2771}, {1291,32423}, {1301,15311}, {1302,14915}, {1303,32428}, {1304,2777}, {1308,2801}, {1309,2818}, {1350,9741}, {1379,3414}, {1380,3413}, {1381,3308}, {1382,3307}, {1670,33707}, {1671,33708}, {2080,8592}, {2222,2800}, {2291,28292}, {2370,32475}, {2373,30209}, {2374,20186}, {2378,27551}, {2379,27550}, {2382,28521}, {2383,20184}, {2384,28294}, {2390,32704}, {2393,30247}, {2687,8674}, {2688,2774}, {2689,2779}, {2690,2772}, {2691,2836}, {2692,2842}, {2693,9033}, {2694,2850}, {2695,2773}, {2696,2854}, {2697,9517}, {2699,2787}, {2700,2786}, {2701,2792}, {2702,2784}, {2703,2783}, {2704,2795}, {2705,2796}, {2706,2797}, {2707,2798}, {2708,2785}, {2710,2799}, {2711,2788}, {2712,2789}, {2713,2790}, {2714,2791}, {2715,2794}, {2716,3738}, {2717,3887}, {2718,2827}, {2719,2828}, {2720,2829}, {2721,2830}, {2722,2831}, {2725,2820}, {2726,2821}, {2727,2822}, {2728,2823}, {2729,2824}, {2730,2835}, {2731,2841}, {2732,2846}, {2733,2849}, {2734,8677}, {2735,2852}, {2736,2809}, {2737,2810}, {2738,2811}, {2739,2812}, {2740,2813}, {2741,9518}, {2743,2802}, {2744,2803}, {2745,2804}, {2746,2805}, {2747,2806}, {2748,9519}, {2749,9520}, {2750,9521}, {2751,2814}, {2752,2775}, {2753,9522}, {2754,9523}, {2755,9524}, {2756,9525}, {2757,2815}, {2758,2776}, {2759,9526}, {2760,9527}, {2761,9528}, {2762,2816}, {2763,9529}, {2764,9530}, {2765,2817}, {2766,2778}, {2767,9531}, {2768,2819}, {2769,9532}, {2770,2780}, {2825,53182}, {2848,53914}, {2867,53795}, {2881,53912}, {3098,7757}, {3357,51877}, {3522,5644}, {3524,5646}, {3534,9716}, {3563,3566}, {3564,3565}, {3800,29180}, {3827,26706}, {3849,6233}, {3880,30236}, {3906,14388}, {3907,29056}, {3910,29206}, {4083,15323}, {4108,45723}, {4132,53905}, {4160,53896}, {4588,28160}, {4715,28318}, {4725,28327}, {4777,28159}, {4778,28193}, {4785,53900}, {4802,28145}, {4912,28320}, {4926,28203}, {4962,28235}, {4971,28305}, {4977,28173}, {5092,31168}, {5237,49952}, {5238,49953}, {5373,5373}, {5544,10304}, {5638,6039}, {5639,6040}, {5643,8703}, {5645,15688}, {5648,7464}, {5652,7418}, {5653,36166}, {5731,47041}, {5762,43344}, {5840,6099}, {5841,43345}, {5842,15439}, {5844,28218}, {5846,28480}, {5847,28477}, {5851,53887}, {5853,53888}, {5854,53897}, {5897,8057}, {5951,8702}, 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