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too complicated to be written here. Click on the link to download a text file. |
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infinite points of K024 |
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Geometric properties : |
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Let X be a finite point. There is one and only one circum-stelloid K(X) with radial center X and asymptotes parallel to those of K024. K(X) is a nK if and only if X lies on a rectangular hyperbola, homothetic and concentric to the Jerabek hyperbola, passing through X(2), X(210), X(381), X(599), X(1853), etc. For instance, K(X2) = K024, K(X599) = K094 and K(X381) = K1137. K(X) is a K0 if and only if X lies on the trilinear polar of X(3228) passing through X(2), X(512), X(3111), X(4108), X(14608). It follows that there are only two K(X) which are nK0 and one of them is K024. The other is K1136 = nK0(X9178, X5968).
The corresponding radial center X of K1136 is the barycentric product X(523) x X(10754), on the lines {2,512}, {381,1499}, {671,690}, etc, SEARCH = 1.11094598783747. See the related bicircular quintic Q153. K024 and K1136 meet again at six finite points on the Kiepert hyperbola, including A, B, C. Hence K1136 belongs to the pencil of cubics generated by K024 and the union of the line at infinity with the Kiepert hyperbola. K1136 meets the circumcircle at the same points as nK0(X6, Q) where Q is on the lines {2,523}, {3,512}, {6,647}, {110,351}, etc, SEARCH = 13.3934743776048. See the related K945 = nK0(X9178, X16092), the orthopivotal cubic O(X5968) with singular focus X(5653). X and Q are X(34290) and X(34291) in ETC (2019-09-18). |