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X(3), X(6), X(40122) infinite points of K003 T1, T2, T3 : vertices of the CircumTangential triangle (T) G1, G2, G3 : vertices of the Grebe triangle (G) other points below |
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Geometric properties : |
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K1161 is the McCay cubic of the Grebe triangle (G) described in page K102. K1161 meets K003 at three infinite points and then six other finite points which lie on a rectangular hyperbola passing through X(i) for i in {3, 22, 32, 251, 3744, 19164}, whose center lies on the lines {5,2794), {25,111}. K1161 is a stelloid with radial center X(5085), the centroid of (G). (T) and (G) are poristic triangles hence there is a conic (C2) inscribed in both triangles. Its center is X(5092) and it passes through X(36213). K1161 meets the sidelines of (G) again at O1, O2, O3 which are the vertices of the cevian triangle of O in this same triangle. Since K is the orthocenter of (G), it is the isopivot of the cubic in (G) hence the tangents at G1, G2, G3 to K1161 concur at K. K1161 meets the sidelines of (T) again at K1, K2, K3 which are the orthogonal projections of K on the sidelines of (T). These points O1, O2, O3, K1, K2, K3 lie on a same conic (C1), concentric to (C2), bitangent to (C2) at two points on the Brocard axis. It follows that (C1) is bicevian in (G) with perspectors O and X(5085). (C2) passes through X(5026) and X(12042). |
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