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X(3), X(98), X(1503), X(1676), X(1677), X(1691), X(2353), X(5000), X(5001), X(5989), X(34129), X(34130), X(34131), X(34132), X(34133), X(34134), X(34135), X(34136)

Ka, Kb, Kc : vertices of the tangential triangle

centers of the 1st Neuberg circles : Oa = X(3)Ka /\ X(98)A, Ob and Oc likewise

other points below

Geometric properties :

K1129 is a circular circumcubic of ABC and KaKbKc. See the analogous K108.

It is a psK with respect to KaKbKc, the pseudo-pivot is Q and the pseudo-isopivot is X(6). Q is the common tangential of Ka, Kb, Kc, on the lines {X2,X160}, {X6,X25}, {X22,X157}, {X23,X385}, {X26,X2353}, etc, SEARCH = -2.28332557171032.

The singular focus F is the inverse X(34217) of X(6033). F lies on the lines {X3,X114}, {X24,X132}, {X112,X186}, etc, SEARCH = 4.82662212897896.

K1134 is the locus of pivots of circular pKs which pass through X(98). The locus of the isopivots is K1129. See K336, K653 for instance.

Other points on K1129 :

• P1 = X(34131), on the circumcircle (C) of KaKbKc, on the lines {X3,X132}, {X22,X107}, {X24,X98}, {X25,X125}, etc, SEARCH = 2.27661064575805.

• P2 = X(34130), isogonal conjugate of X(147), on the line {X147,X325}, SEARCH = 10.4017337681930.

• P3 = X(34129), the tangential of X(1503) hence on the real asymptote, on the lines {X3,X132}, {X127,X2207}, etc, SEARCH = 1.81044345670380.

• P4, the tangential of X(1691), on the line {X2353,X5989}, SEARCH = 3.35576376837979.

Note that the tangential of X(3) is X(1691).

A remarkable property :

The inversive image iK1129 of K1129 in (O) and cgK1129, the complement of its isogonal transform, coincide with K570 = psK(X237, X2, X3), see CL068. Hence K1129 is stable under the mappings icg and gai with the usual notations. For instance, P5 and P6 are two other points on K1129 :

• P5 = icgX(5989) =X(34132), the tangential of X(98), SEARCH = 0.269861240273307.

• P6 = icgP4, on the lines {X1691,X2353}, {X5989,P3}, etc, SEARCH = 0.904599406048610.

Triads of collinear points on the cubic : {X3, X1503, X2353} – {X3, P1, P3} – {X3, P2, P5} – {X98, X1503, X1691} – {X98, X2353, P1} – {X1503, X1676, X1677} – {X1691, X2353, P6} – {X1691, P1, P2} – {P3, P4, P5}.

Note that the locus of M such that X(3), M, icgM are collinear is the Lemoine cubic K009 (together with the circumcircle).

The isogonal transform of K1129 is K1131.