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IGOHK cubics

The Thomson cubic has the remarkable property to contain five of the most common centers of the triangle namely I, G, O, H, K. We study all the other circum-cubics sharing the same property. See the related CL061 and CL062.

Proposition 1 : All the circum-cubics passing through I, G, O, H, K form a pencil of cubics which is stable under isogonality. When they are not singular (see below), they are all tangent at I to the line IG.

In other words, the isogonal transform of any such cubic is another cubic of the same type.

If K(P) denotes the cubic of the pencil which contains P, then K(P*) is its isogonal transform. If they are distinct, these two cubics generate the pencil.

Proposition 2 : this pencil contains two self-isogonal cubics. These are the Thomson cubic (the only pK of the pencil) and the Thomson cK sister K383 (the only non degenerate nK of the pencil, a nodal cubic).

Note that a cubic of this type cannot be circular nor equilateral.

It is clear that K(P) decomposes when it contains a fourth point on the Euler line or a seventh point on the Jerabek hyperbola. In the former case, K(P) is the union K1 of the Euler line and the circum-conic through I and K. In the latter case, K(P) is the union K2 of the Jerabek hyperbola and the line IG. These two decomposed cubics generate the pencil i.e. it is always possible to find two real numbers k1, k2 such that, for any point P, we can write K(P) = k1 K1 + k2 K2 = K(k1, k2) and K(P*) = k2 K1 + k1 K2 = K(k2, k1) with suitably normalized equations for K1, K2.

When k1 and k2 are equal or opposite, we obtain the two self-isogonal cubics as seen above.

The following table gives a selection of such cubics.

P

K(P) contains I, G, O, H, K and ...

cubic

P, the third point on OK

 

X15

X14, X15, X559, X1251

 

X16

X13, X16, X1082

 

X32

X32, X83, X1429, X2344

 

X39

X39, X76

 

X61

X18, X61

 

X62

X17, X62

 

X182

X98, X171, X182, X983

 

X187

X187, X598

 

X216

X216, X2052

 

X284

X81, X284, X1172, X1751, X1780

 

X371

X371, X486

 

X372

X372, X485

 

X500

X79, X500

 

X511

X256, X262, X511, X982

 

X566

X94, X566

 

X569

X96, X569

 

X572

X572, X940, X2221, X2298

 

X573

X573, X941, X2051, X2345

 

X574

X574, X671

 

X577

X275, X577

 

X579

X37, X226, X579, X1214, X2335

 

X580

X580, X943

 

X581

X581, X942

 

X582

X35, X582

 

X970

X970, X986

 

X991

X7, X354, X672, X955, X991

K382 = ETC cubic

X1151

X1132, X1151

 

X1152

X1131, X1152

 

P, the third point on GK

 

X81

X81, X284, X1172, X1751, X1780

 

X193

X193, X2271

 

X394

X219, X394, X2982

 

X940

X572, X940, X2221, X2298

 

X3945

X3945

K383 = Thomson cK sister

P, the third point on HK

 

X393

X278, X393

 

X1172

X81, X284, X1172, X1751, X1780

 

X1249

X9, X57, X223, X282, X1073, X1249

K002 = Thomson

X1498

X1433, X1498

 

P, the sixth point on Kiepert

 

X13

X13, X16, X1082

 

X14

X14, X15, X559, X1251

 

X17

X17, X62

 

X18

X18, X61

 

X76

X39, X76

 

X83

X32, X83, X1429, X2344

 

X94

X94, X566

 

X96

X96, X569

 

X98

X98, X171, X182, X983

 

X226

X37, X226, X579, X1214, X2335

 

X262

X256, X262, X511, X982

 

X275

X275, X577

 

X485

X372, X485

 

X486

X371, X486

 

X598

X187, X598

 

X671

X574, X671

 

X1131

X1131, X1152

 

X1132

X1132, X1151

 

X1751

X81, X284, X1172, X1751, X1780

 

X2009

X1689, X2009

 

X2010

X1690, X2010

 

X2051

X573, X941, X2051, X2345

 

X2052

X216, X2052

 

P, the sixth point on Feuerbach

 

X7

X7, X354, X672, X955, X991

 

X9

X9, X57, X223, X282, X1073, X1249

K002 = Thomson

X79

X79, X500

 

X90

X90, X1728

 

X104

X104, X999

 

X256

X256, X262, X511, X982

 

X294

X218, X294, X949

 

X941

X573, X941, X2051, X2345

 

X943

X580, X943

 

X983

X98, X171, X182, X983

 

X1000

X517, X1000

 

X1172

X81, X284, X1172, X1751, X1780

 

P, the third point on IO

 

X35

X35, X582

 

X55

X55, X673, X954, X2346

 

X57

X9, X57, X223, X282, X1073, X1249

K002 = Thomson

X171

X98, X171, X182, X983

 

X241

X241, X277, X948

 

X354

X7, X354, X672, X955, X991

K382 = ETC cubic

X517

X517, X1000

 

X559

X14, X15, X559, X1251

 

X940

X572, X940, X2221, X2298

 

X942

X581, X942

 

X982

X256, X262, X511, X982

 

X986

X970, X986

 

X999

X104, X999

 

X1214

X37, X226, X579, X1214, X2335

 

X1319

X1319, X2320

 

P, the third point on IH

 

X223

X9, X57, X223, X282, X1073, X1249

K002 = Thomson

X226

X37, X226, X579, X1214, X2335

 

X278

X278, X393

 

X581

X581, X942

 

X948

X241, X277, X948

 

X1457

X392, X957, X1457

 

P, the third point on IK

 

X9

X9, X57, X223, X282, X1073, X1249

K002 = Thomson

X37

X37, X226, X579, X1214, X2335

 

X44

X44, X89

 

X45

X45, X88

 

X218

X218, X294, X949

 

X219

X219, X394, X2982

 

X220

X220, X1170, X2338

 

X238

X238, X985

 

X392

X392, X957, X1457

 

X518

X518, X1002

 

X954

X55, X673, X954, X2346

 

X958

X958, X961

 

X960

X959, X960

 

X984

X291, X984

 

X1001

X105, X1001, X1617

 

X1107

X330, X1107

 

X1124

X1124, X2066

 

X1212

X279, X1212

 

X1728

X90, X1728

 

X2176

X1258, X2176

 

 

GOHK cubics

Isogonal GOHK cubics

Since G, K and O, H are two pairs of isogonal conjugates, there is only one isogonal pK passing through G, O, H, K : this is the Thomson cubic.

***

Let us now consider an isogonal nK passing through G, O, H, K. We know that the tangents at two isogonal points on the cubic meet at the isogonal conjugate of the third point of the cubic on the line through the two initial points. Hence the tangents at O and H must meet at K. This show that all isogonal nKs pass through seven fixed points (A, B, C, G, O, H, K) and have two fixed tangents at O, H. They form a pencil of nKs which can be generated by any two of them.

In particular, there are three decomposed cubics in the pencil :

- the union of the Euler line and the Jerabek hyperbola,
- the union of the Brocard line and the Kiepert hyperbola,
- the union of the line HK and the circumconic through G, O.

The roots of all these nKs lie on the line passing through X(648), X(110), X(107) which are the roots of the three decomposed cubics.

Any nK of this type is the locus of M such that

(1) M and its isogonal conjugate M* are conjugated with respect to a fixed circle which must have its center on the radical axis of the circles with diameters GK and OH. This is the line passing through X(74) and X(98).
(2) the pedal triangle of M (and M*) is orthogonal to a fixed circle. This circle must have its center on the trilinear polar of X(523) i.e. the line through X(115), X(125), etc.

If another point P of the cubic is given, the circle in (1) is centered at the radical center of the three circles with diameters GK, OH, PP* and is orthogonal to these circles.

The table gives a selection of these isogonal nKs passing through a given point P.

P

the cubic contains G, O, H, K and...

cubic

X1

X1 (focal cubic)

K072

X7

X7, X55, X672, X673, X942, X943

K385

X8

X8, X56, X104, X517, X1193, X1220

K386

X9

X9, X40, X57, X84

K384

X19

X19, X63, X2285, X2339

 

X31

X31, X75, X2276

 

X44

X44, X45, X88, X89

 

X88

X44, X45, X88, X89

 

X105

X105, X518, X1001, X1002

 

X106

X106, X519, X995, X996

 

X145

X145, X1201, X1222

 

X171

X171, X256, X986, X987

 

X184

X184, X232, X264, X287

 

X185

X185, X1092, X1093, X1105

 

X192

X192, X1575, X2162

 

X195

X195, X1157, X1263

 

X200

X200, X269, X2297, X2999

 

X219

X219, X278, X2982

 

X220

X220, X279, X1170, X1212

 

X222

X222, X281, X1465

 

X223

X223, X282, X1490

 

Other GOHK cubics

Let us consider the transformation f which maps a point M to f(M) = M#, the intersection of the lines GM and KM* (M* is the isogonal conjugate of M). f is very similar to the Cundy-Parry transformations as seen in CL037.

f is a third degree involution with singular points A, B, C, K, G (double). Its fixed points are those of the Grebe cubic.

f transforms any circum-cubic (K') passing through G and K into another cubic (K") of the same type. (K') meets the Grebe cubic at A, B, C, G, K and four other points which are fixed under f. This shows that (K') and (K") have nine known common points and generate a pencil of cubics containing the Grebe cubic. Recall that this pencil is invariant under isogonal conjugation.

f swaps O and H hence f transforms any cubic through G, O, H, K into itself. This gives the

Proposition 3 : any circum-cubic (K) passing through G, O, H, K is (globally) fixed by f.

Consequences :

  1. if (K) contains another given point P, it must contain f(P) = P# = GP /\ KP*.
  2. (K) meets the Grebe cubic at A, B, C, G, K and four other points which must lie on the polar conic PC(G) of G in (K). This polar conic always contains G, L = X(20) and X(194).
  3. (K) and the Thomson cubic have already seven common points (A, B, C, G, O, H, K) and must meet at two other points. These two points are isogonal conjugates and are collinear with G. Indeed, f swaps any two isogonal conjugates on the Thomson cubic. When these two points are X(9) and X(57), we obtain a pencil of cubics and several examples are given in the table below. Each cubic has five known common points with the Kiepert hyperbola (A, B, C, G, H) and with the Feuerbach hyperbola (A, B, C, G, X9). The sixth points on each hyperbola are the green and red points respectively.

P

the cubic contains G, O, H, K, X9, X57 and ...

cubic

X1

X1, X223, X282, X1073, X1249

K002

X8

X8, X392, X957, X1193, X2183, X3057

K387

X10

X10, X37, X386

 

X13

X13, X16, X1277

 

X14

X14, X15, X1276, X2306

 

X32

X32, X83, X238, X985, X987

 

X39

X39, X76, X291, X984, X986

 

X40

X40, X84

K384

X56

X56, X956, X1220, X1476

isogonal of K387

X79

X79, X2245, X2895

 

X98

X98, X182 , X2329

 

X103

X103, X220 , X1170

 

X104

X104, X572, X958 , X961, X1150, X1766

 

X105

X105, X169, X943, X1001

 

X219

X219, X580, X2982

 

X262

X262, X511, X1432, X3061

 

X275

X275, X577, X3074

 

X279

X279, X516, X991, X1212

 

X281

X281, X393, X1465

 

X372

X372, X485, X2362

 

X517

X517, X573, X959, X960, X2051

 

X518

X518, X942, X1002

 

 

 

 

Coming back to the general case, it seems convenient to characterize (K) with two points P1, P2 on the lines OK, GK respectively. In this case, P1# = f(P1) is the second intersection of the line GP1 and the Kiepert hyperbola. (K) meets the line P1P2 again at P3 on the rectangular circum-hyperbola passing through P2 and obviously P3# = f(P3) is another point on (K). Since we now have four collinearities (GOH, GP1P1#, GKP2, GP3P3#), it is easy to construct PC(G). This also gives the tangent at G to (K) and the tangential G+ of G (on the circumconic through G and K). The tangential K+ of K is the intersection of the lines HP3 and KP2*. Note that the coresidual P4 of A, B, C, G and P4# are two other points on (K) : P4 is the intersection of the lines HP1# and KG+.

***

Construction of (K) with given P1, P2 on the lines OK, GK respectively.

A variable line (L) passing through G meets PC(G) at N and the Grebe cubic at two points lying on the circum-conic (C) which is the K-Ceva conjugate of (L). These two points are not always real and we shall not try to use them in the construction. These points are in fact the fixed points of the involution on (L) which swaps a point M on (L) and the intersection M' of (L) with the polar line of M in (C). Note that (C) contains the vertices of the cevian triangle of K, X(194) = K-Ceva conjugate of G and the K-Ceva conjugate of the infinite point of (L). This latter point lies on the bicevian conic C(G,K).

Hence we can construct G' and N' on (L) as above. The circle centered on (L) which is orthogonal to the circles with diameters GN and G'N' meets (L) at two points on (K).

table30fig1