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X(69), X(75), X(264), X(18749)

vertices of triangles ABC, cevian-of-X(69) = pedal-of-X(20), anticevian-of-X(75)

Geometric properties :

K1399 is the isotomic transform of the McCay cubic K003. It is a K+ with three real asymptotes concurring at X = b^2 c^2 (-3 a^6 b^2+4 a^4 b^4-a^2 b^6-3 a^6 c^2+7 a^4 b^2 c^2+a^2 b^4 c^2+b^6 c^2+4 a^4 c^4+a^2 b^2 c^4-2 b^4 c^4-a^2 c^6+b^2 c^6) : : , on the line {53,141}.

Locus property

(contributed by César Lozada, 2025-05-14)

Let P be a point with cevian triangle PaPbPc. The cevian lines of P meet the Steiner ellipse again at Qa, Qb, Qc respectively.

The inconic with perspector Qa is a parabola P(A) tangent at Pa to the line BC, and obviously also tangent to the lines AB, AC. Two other parabolas P(B) and P(C) are defined cyclically.

The axes of these three parabolas are concurrent (at Q) if and only if P lies on K1399. In this case, Q = atg(cP ÷ P), where cP ÷ P is the barycentric quotient of cP and P.

Some  ETC pairs (P, Q) : (69, 20), (75, 1), (264, 5889).

Other points on K1399 : 

P1 = X(68534) = tX1745 = cross conjugate of {X264, X75}, on the lines {69, 18749}, {332, 3362}, {333, 18751}, {345, 7361}, {7049, 30479}, {7182, 46752}, {20930, 57801}, {24031, 57806}, {44130, 60801}.

P2 = X(68535) = tX13855 = X(69)-Ceva conjugate of X(264), on the lines {5, 264}, {69, 56271}, {92, 18161}, {95, 40800}, {317, 1899}, {343, 15466}, {3964, 6331}, {6528, 20477}, {19211, 43752}, {33808, 57812}, {44133, 56593}, {52581, 57909}.

P3 = X(69)X(56271) /\ X(394)X(8613).