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X(6), X(115), X(187), X(524), X(5526), X(9721), X(16303), X(28662), X(44909), X(48653), X(48654), X(48721), X(50536)

X(39162), X(39163), X(39164), X(39165) : foci of the Steiner inellipse

projections of K on the sidelines of the Grebe triangle

Geometric properties :

K1275 is another strophoid related to K018 and very similar to K1274.

Its singular focus is the midpoint X(28662) of {X6,X111} and its node is K. The nodal tangents are the axes of the orthic inconic (K) and they are parallel to the asymptotes of the Jerabek hyperbola (J).

Its real asymptote passes through X(126) and X(524). It is parallel to the orthic line of K1275 which is GK. These two lines are the same for K1274.

The polar conic of K in K018 is the rectangular hyperbola (H) with center G, passing through X(3), X(6), X(381), X(599), X(2574), X(2575) hence homothetic to (J).

K1275 is the Psi2-image of (H), where Psi2 is the transformation described in pages K018, K1142 and related to the stelloid K598.

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A line passing through K meets K018 again at two points P1, P2 which lie on a circum-conic passing through G. Recall that K018 is an isogonal cubic in the triangle GP1P2.

The barycentric product P1 x P2 lies on K381 = cK(#X6, X523).

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

The midpoint Q of P1, P2 lies on K1275.

In particular, K1275 passes through the midpoints of {A,A2}, {B,B2}, {C,C2} where A2B2C2 is the second Brocard triangle.

Two perpendicular lines passing through K correspond to two points Q, Q' on K1275 which are collinear with the singular focus X(28662).