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K1153

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X(2), X(671), X(6094), X(11054), X(17948), X(22329), X(37854), X(37855), X(37856), X(37857), X(37858), X(37859), X(37860), X(38888), X(38889), X(38890), X(38891), X(38892), X(38893), X(38894), X(38895)

Geometric properties :

Let G, K be the points X(2), X(6) in ETC, Q another fixed point on the circumcircle (O) of ABC and M a variable point on the line GK. The circumcircle of GQM meets the circum-conic of ABC passing through G and M at G, M and two other points M1, M2. When M traverses GK, the locus of M1, M2 is a circular circum-cubic K(Q) = nK(Ω, R, G) passing through G, X(523), Q and its pole Ω which is the barycentric product Q x X(523). For any point M, the line M1M2 passes through the pole Ω.

It follows that Ω must lie on the circum-hyperbola (C) passing through G, K and many other centers listed at the bottom of this page. Note that Q, Ω and X(111) are collinear.

When Q varies on (O), G, Ω, R and X(22329), Q, R are also two triads of collinear points and the locus of R is the nodal cubic K1153 with node G and nodal tangents parallel to the asymptotes of (C).

For instance, with Q = X(99), we obtain the isotomic nodal K(Q) = K088 = nK(X2, X11054, X2) = cK(#X2, X11054) and with Q = X(110), we obtain the isogonal focal K(Q) = nK(X6, X22329, X2).

The singular focus F of K(Q) lies on a hyperbola passing through X(3), X(110), X(14653).

K1153 is a member of the pencil of cubics generated by K185 = cK(#X2, X523) and the union of the Kiepert hyperbola (K) and the line through G and X(523). This pencil contains several other decomposed cubics namely :

• the union of the circum-conic passing through G, X(892), X(17948) and the line through G and X(99).

• thrre cubics, each being the union of a median of ABC and the conic passing through G, X(671), X(17948) and the two remaining vertices of ABC.

K1153a

K1153 meets the Steiner ellipse (S) at A, B, C, X(671) and two imaginary points on the line (L) passing through X(351), X(523) and X(22329) which lies on K1153.

K1153 meets (C) at A, B, C, G (counted twice) and X(6094). See below.

If M = u : v: w is a point different from G then the following point N on the line GM also lies on K1153. Its first coordinate is :

(a^2 + b^2 + c^2)^2 - 6 b^2 c^2 + 3 a^2 (b^2 u - c^2 u - b^2 v + c^2 w) / (v - w)

the other coordinates are obtained cyclically.

Obviously, M = N when M is a point on K1153.

For instance, when Q = X(9080) = tripole of the line {6, 543}, K(Q) is nK(X6094, X6094, X2).

***

Annexe : centers X(i) on the hyperbola (C) for these i :

2, 6, 25, 37, 42, 111, 251, 263, 308, 393, 493, 494, 588, 589, 694, 941, 967, 1169, 1171, 1218, 1239, 1241, 1383, 1400, 1427, 1880, 1976, 1989, 2054, 2165, 2248, 2350, 2395, 2433, 2963, 2981, 2987, 2998, 3108, 3228, 3444, 3457, 3458, 3572, 5638, 5639, 6094, 6096, 6151, 6339, 8105, 8106, 8576, 8577, 8749, 8770, 8791, 8794, 8882, 9178, 9281, 9403, 9462, 10103, 11166, 11175, 13854, 14553, 14579, 14606, 14624, 14840, 14842, 14910, 14948, 14998, 16081, 16098, 16606, 18372, 18818, 18898, 21448, 21461, 21462, 24861, 27809, 28625, 28658, 30535, 30537, 33631, 34079, 34204, 34212, 34288, 34533, 34534, 34570, 34572, 34816, 34818, 34898, 36616, 37128.

Recall that (C) is the circum-conic with perspector X(512) and center X(1084).