   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 : if (K) contains another given point P, it must contain f(P) = P# = GP /\ KP*. (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). (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 blue 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).        