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X(3), X(4), X(99), X(512), X(7468), X(14265), X(34157), X(51480)

imaginary foci of the MacBeath inconic

other points below

Geometric properties :

K1282 is a spK as in CL055. See Table 73 for properties and other related cubics.

It is an isogonal focal non-pivotal isocubic, namely nK(X6, R, X3) as in CL062. Its singular focus is X(99), Steiner point on the circumcircle.

The root R is the barycentric product X(76) x X(230), on the trilinear polar of X(264) and on the lines {X2, X39}, {X4, X52}, {X6, X311}, {X94, X671}, {X110, X419}, {X141, X1232}, SEARCH = 0.657131568250371. R is the trilinear pole of the line {X114, X2974} which meets the sidelines of ABC at three points A', B', C' on the cubic. R is now X(51481) in ETC.

The orthic line (L) is {X5, X512} which meets the cubic at one real point X(512) and two imaginary points on K005 and on the circum-conic passing through X(54), X(99). This is the isogonal transform of (L).

These two points also lie on the polar conic (C) of X(99), a circle passing through X(99) and X(2142).

X(99) and X(512) share the same tangential which is the point X where K1282 meets its asymptote, the perpendicular at X(6321) to the Brocard axis. X(6321) is the reflection of X(99) in X(5).

The isogonal conjugate Y of X is the third point of the cubic on the perpendicular at X(99) to the Brocard axis.

Q1 = {99,7468} /\ {512,51480} and Q2 = {512,7468} /\ {99,51480} are two other isogonal conjugates on the cubic, now X(51479) and X(51478) in ETC.

Locus properties

K1282 is the locus of

• foci of inconics with center on the line (L) = {X5, X512}.

• contacts of tangents drawn through X(99) to the circles of the pencil with radical axis (L) which contains (C).

• P such that the directed line angles (PO, PX99) and (PH, PX512) are opposite (mod.π).

• P such that its isogonal conjugate and its reflection in the Euler line are collinear with X(7468), the third point of K1282 on the Euler line.

• common points of a parallel to (L) and the isogonal transform of its reflection about X(5) or about (L). Hence, two lines (L1), (L2), symmetric about (L), and the corresponding conics (C1), (C2), give two pairs of isogonal conjugate points, two by two collinear with X(99). See figure below.