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K1274

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X(2), X(125), X(524), X(468), X(5205), X(6108), X(6109), X(9172), X(46980), X(47638)

X(48439) → X(48444)

foci of the orthic inconic

projections of G on the sidelines of the Thomson triangle

see points below

Geometric properties :

K1274 is a strophoid with singular focus X(9172) and node G. The nodal tangents are parallel to the asymptotes of the Kiepert hyperbola (K) and to the axes of the Steiner inelllipse (S).

Its real asymptote passes through X(126) and X(524). It is parallel to the orthic line of K1274 which is GK.

K1274 is the inverse of (K) with respect to the orthoptic circle of (S). X (tangential of X524 and X9172 on the asymptote) and Y (on its parallel at X9172) are the inverses of X(43667) and X(43674) respectively.

K1274 is also the Psi-image of the polar conic (H) of G in K018. (H) is the rectangular hyperbola with center K that passes through X(2), X(4), X(194), X(1689), X(1690), X(1992), X(3413), X(3414) hence homothetic to (K). (H) also contains the foci of the orthic inconic. Its equation is : (b^2 - c^2)(SA x^2 + a^2 y z) = 0.

K1274a

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A line passing through G meets K018 again at two points P1, P2 which lie on a circum-conic passing through K.

The barycentric product P1 x P2 lies on K222 = cK(#X6, X512).

The harmonic conjugate of G in P1, P2 lies on (H).

The midpoint Q of P1, P2 lies on K1274.

In particular, K1274 passes through the midpoints of {A,A4}, {B,B4}, {C,C4} where A4B4C4 is the fourth Brocard triangle (dashed blue triangle called 5th Euler triangle in ETC, see preamble to X3758).

Two perpendicular lines passing through G correspond to two points Q, Q' on K1274 which are collinear with the singular focus X(9172).

Other inverses of points on (K) :

X76 : a^6 b^4 + a^4 b^6 - 4 a^6 b^2 c^2 - 3 a^2 b^6 c^2 + a^6 c^4 + 2 a^2 b^4 c^4 + 2 b^6 c^4 + a^4 c^6 - 3 a^2 b^2 c^6 + 2 b^4 c^6 : : = X(48439), SEARCH = 7.482334533200099

X83 : (b^2 + c^2) (-4 a^8 - 3 a^6 b^2 + b^8 - 3 a^6 c^2 + 2 a^4 b^2 c^2 + 3 a^2 b^4 c^2 - b^6 c^2 + 3 a^2 b^2 c^4 + 2 b^4 c^4 - b^2 c^6 + c^8) : : = X(48440), SEARCH = -16.3070481427064

X262 : a^2 (a^4 b^4 - a^2 b^6 + 4 a^4 b^2 c^2 - 2 a^2 b^4 c^2 + b^6 c^2 + a^4 c^4 - 2 a^2 b^2 c^4 - 2 b^4 c^4 - a^2 c^6 + b^2 c^6) : : = X(47638), SEARCH = -1.516663306478879

X1916 : a^6 b^4 + 2 a^4 b^6 - 4 a^6 b^2 c^2 + a^4 b^4 c^2 - 3 a^2 b^6 c^2 + a^6 c^4 + a^4 b^2 c^4 + b^6 c^4 + 2 a^4 c^6 - 3 a^2 b^2 c^6 + b^4 c^6 : : = X(48444), SEARCH = 1.50528215110294