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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.

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.

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 blue and red points respectively.

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).