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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 1st 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 2nd 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 T are perspective at P. (César Lozada, ADGEOM #1168).

• If T is the 1st Sharygin triangle, the locus of P is K960 = pK(X1914, X8424), the blue curve on the figure, with X(8424) = X(6)X(256) /\ X(7)X(21) /\ X(75)X(1281), etc.

• If T is the 2nd Sharygin triangle, the locus of P is K961 = pK(X1914, X8301), the green curve on the figure, with X(8301) = X(1)X(1929) /\ X(2)X(11) /\ X(75)X(1281), etc.

See also K132 and K323 for analogous properties with the cevian triangle of M.

K673 is the isogonal transform of K136.