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X(1), X(2), X(190), X(513)

infinite points of the Steiner ellipse

a1 = a : c : b, b1 = c : b : a, c1 = b : a : c

a3 = bc : b^2 : c^2, b3 = a^2 : ac : c^2, c3 = a^2 : b^2 : ab

The parallel at I to BC meets the A-cevian of X(75) at a1. The I-isoconjugate of a1 is a3. Define b1, c1, b3, c3 similarly. Then,

  • the triangles ABC, a1b1c1 are triply perspective at X(75), J1 = ac : ba : cb and J2 = ab : bc : ca (Jerabek points),
  • the triangles ABC, a3b3c3 are triply perspective at X(6), J1, J2,
  • the triangles a1b1c1, a3b3c3 are triply perspective at X(894), J1, J2.

For any point P, let Pa = a1P /\ BC, Pb = b1P /\ CA, Pc = c1P /\ AB (or similarly with a3b3c3). Pa, Pb, Pc are collinear if and only if P lies on K324. The triangles ABC and a1b1c1 (or a3b3c3) are perspective if and only if P lies on K132. See a generalization at CL041.

K324 meets the line at infinity at the same points as the Steiner ellipse and has only one real asymptote (the line X291-X513).

The isogonal transform of K324 is nK0(X31, X238) and its isotomic transform is nK0(X75, X350).

K324a

K324 has always three real prehessians P1, P2, P3.

The centers of the polar conics of X(513) with respect to these prehessians are X(1), X(2), X(190). It follows that K324 is a pK with pivot X(513) with respect to the triangle X(1)X(2)X(190).

The isopivot is the point X on the real asymptote. X is the intersection of the lines X(1)X(659), X(2)X(812).

The polar conic of X contains X(1), X(2), X(37), X(190), X(513).

Thus K324 must contain the vertices of the diagonal triangle of the quadrilateral X(1), X(2), X(190), X(513).