Home page | Catalogue | Classes | Tables | Glossary | Notations | Links | Bibliography | Thanks | Downloads | Related Curves

∑ (b^2 + c^2 - 2a^2) x (c^2y^2 - b^2z^2) = 0

X(1), X(2), X(6), X(111), X(524), X(2930), X(5524), X(5525), X(7312), X(7313), X(8591), X(13574), X(52678), X(52679)

excenters

Geometric properties :

K1156 is the isogonal circular pK with pivot X(524) and singular focus X(1296).

K1156 is a member of the Thomson-Grebe pencil. See Table 13.

Locus property

Let P be a point and denote by Oa, Ob, Oc the circumcenters of PBC, PCA, PAB and by Ka, Kb, Kc the symmedian points of PObOc, POcOa, POaOb. These triangles OaObOc and KaKbKc are perspective if and only if P lies on K1156 (Kadir Altintas). Recall that ABC and OaObOc are perspective if and only if P lies on the Neuberg cubic K001 (see locus property 2).

Another related pencil of cubics

K018 = nK0(X6, X523) and K1156 = pK(X6, X524) are two isogonal circular cubics that meet at nine identified points, namely A, B, C, X(2), X(6), X(111), X(524) and the circular points at infinity. They generate a pencil of circular circum-cubics K0 (no term in xyz), passing through these same points, which is stable under isogonal conjugation.

If T is a real number, each cubic can be written K(T) = (1 - T) K018 + T K1156. Thus, K(0) = K018 and K(1) = K1156.

The singular focus F(T) of every cubic lies on the line X(3), X(111), X(1296), etc.

K(T) meets its real asymptote at X(T) which lies on the circum-hyperbola passing through X(111), X(524), X(6088).

This pencil also contains :

• two decomposed cubics, which are actually the two other nK0s of the pencil :

– K(∞), the union of the line at infinity and the circum-conic with perspector X(512), passing through X(2), X(6), X(25), X(37), X(42), X(111), X(251), and many other points.

– K(1/2), the union of the line (GK) and the circum-circle.

• two other pKs which are K043 = pK(X187, X2) = K(1/4) and its isogonal transform K273 = pK(X111, X671) = K(-1/2).

• one axial cubic whose singular focus lies on the perpendicular bisector of GK.

• two complicated focal cubics which are real when ABC is acute. Each is the isogonal transform of the other.

• one K+ whose singular focus lies on the line G, X(187).

Properties :

• the isogonal transform of K(T) is K(T / (2T-1)). Their singular foci are inverse in (O).

• F(T) and F(1-T) are symmetric about O.

• F(T) and F(-T) are symmetric about X(111).

• F(T) and F(1/T) are inverse in the circle with center X(111), radius 2R.

Points on the sidelines of ABC :

These points U, V, W are defined as follows. The trilinear polars L(X99) and L(X523) meet the sidelines of ABC at Pa, Pb, Pc and Qa, Qb, Qc. L(X99) is the line (G,K) and L(X523) is the line (X115,X125). Note that the midpoint of PxQx is that of the corresponding side of ABC.

U is the homothetic of Pa under h(Qa, 2T), V and W likewise.

Points on the Brocard axis :

These points are X(6) and two other (always real) points which are inverse in the imaginary circle with center X(6) and radius i √3 tanω R, where ω is the Brocard angle.

This circle has equation : ∑ b^2c^2x^2 + a^2 SA y z = 0.

Examples of pairs of such points : {3,187}, {6,511}, {15,16}, {32,35002}, {39,9301}, {182,5104}, {574,2080}, {576,8586}, {1326,37508}, {1350,2030}, {1351,5107}, {1379,1380}, {1384,18860}, {1570,44456}, {1691,3098}, {1692,33878}, {2028,38596}, {2029,38597}, {2076,5092}, {2459,6200}, {2460,6396}, {3003,15919}, {3581,32761}, {5008,47618}, {5033,35383}, {5111,37517}, {5162,26316}, {5164,51340}, {5210,47113}, {6221,6566}, {6398,6567}, {8588,38225}, {10645,39554}, {10646,39555}, {11477,44496}, {14538,41407}, {14539,41406}, {15544,37496}, {19780,36756}, {19781,36755}, {31884,38010}, {37483,50387}, {39229,39230}, {41413,47619}, {51206,51207}.

Points on the Steiner ellipse :

K(T) meets the Steiner ellipse at six points (including A, B, C) that lie on an isotomic pK with pivot S(T) on the line {99, 187, 385, 538, etc}.

S(T) = 2 (a^2 b^2+a^2 c^2-2 b^2 c^2) T - (a^2-b c) (a^2+b c) : : .

The three remaining common points lie on a line passing through G, one of them being G itself.

***

The following table, contributed by Peter Moses, gives a selection of cubics passing through at least six ETC centers.

Knnn* is the isogonal transform of Knnn.