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K959

∑ (b^2 + c^2) x^2 (c^2 SB y – b^2 SC z) = 0

X(2), X(22), X(25), X(251), X(1799), X(8793), X(8879), X(8880), X(8881), X(13575)

X(16277) = X(3313)* = isogonal conjugate of X(3313), on the Kiepert hyperbola

A'B'C' = cevian triangle of X(1799)

(yellow) circumcevian triangle of X(2)

cevian triangle of X(251) wrt circumcevian triangle of X(2)

cevian triangle of X(25) wrt cevian triangle of X(1799)

other points below

Geometric properties :

K959 is the isogonal transform of K140, also the barycentric quotient of K140 by X(141).

K959 meets the line at infinity at the same points as (K1) = pK(X2, P1) where P1 = X(16275) = X(2)X(187) /\ X(305)X(315). The six remaining common points are A, B, C, X(2), X(1799), X(13575) on the circum-conic with perspector X(525).

K959 meets the Steiner ellipse at the same points as (K2) = pK(X2, P2) where P2 = X(16276) = X(2)X(99) /\ X(22)X(76). The three remaining common points are those on the Euler line namely X(2), X(22), X(25).

Note that the line P1P2 is parallel to the Euler line.

X(25) is the isopivot of K959 whose polar conic is a remarkable circum-hyperbola passing through X(i) for these i : 3, 25, 32, 98, 184, 228, 878, 1402, 1410, 1799, 2200, 2351, 2353, 3425, 3437, 3438, 3439, 3442, 3443, 3455, 3456, 3504, 6401, 6402, 8825, 8858, 8884, 10547, 14600, 14908.

Its equation is ∑ a^4 (b^2 - c^2) (-a^2 + b^2 + c^2) y z = 0, that of the circum-conic with perspector X(3049).

Locus properties

  1. locus of P whose cevian triangle is perspective (at Q) to the Ara triangle (see ETC, X5594). Q lies on K174.
  2. the circumcevian triangle of X(2) and the cevian triangle of P are perspective (at Q) if and only if P lies on the cubic pK(X308, X308), a member of CL007, the isotomic transform of K836. Q lies on K959.
  3. the circumcevian triangle of X(2) and the anticevian triangle of P are perspective (at Q) if and only if P lies on K644. Q lies on K959 which is a circumcevian pK. See CL072.