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K673

a^2(a^2 - bc) (c^2 y - b^2 z) y z = 0

X(1), X(6), X(43), X(81), X(238), X(239), X(256), X(291), X(294), X(1580), X(2068), X(2069), X(2238), X(2665)

vertices of the cevian triangle of X(6)

The first Sharygin triangle is the triangle bounded by the perpendicular bisectors of the segments APa, BPb, CPc where Pa, Pb, Pc are the vertices of the incentral triangle (cevian triangle of X(1)).

The second Sharygin triangle is the triangle bounded by the perpendicular bisectors of the segments AQa, BQb, CQc where Qa, Qb, Qc are the traces of the antiorthic axis (trilinear polar of X(1)).

K673 is the locus of M such that the anticevian triangle of M and one Sharygin triangle are perspective at P. (César Lozada, ADGEOM #1168).

In the first case, the locus of P is pK(X1914, P1), the blue curve on the figure, with P1 = X(8424) = X(6)X(256) /\ X(7)X(21) /\ X(75)X(1281).

This cubic contains X(21), X(256), X(846), X(1281), X(1284), X(1580).

In the second case, the locus of P is pK(X1914, P2), the green curve on the figure, with P2 = X(8301) = X(1)X(1929) /\ X(2)X(11) /\ X(75)X(1281).

This cubic contains X(105), X(291), X(1281), X(1282), X(1929), X(2108).

K673 is the isogonal transform of K136.