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K851

too complicated to be written here. Click on the link to download a text file.

X(3), X(4), X(6), X(376), X(15258)

infinite points of K002

points of K243 = pK(X6, X376) on (O)

foci of the inellipse with center X(3)

isogonal conjugates of the X3-OAP points, see Table 53

other points below

K851 is spK(X2, X3) as in CL055. See also Table 54.

Its isogonal transform is K810.

K851 meets the sideline BC at A' which lies on the parallel at X(376) to the median AG. B' and C' are defined similarly.

K851 meets the altitude HA at Ha on the line OA' and the symmedian KA at Ka on the line X(376)A'. Hb, Hc and Kb, Kc are defined similarly.

If Oa, Ob, Oc are the third points on the cevian lines of O, the lines OaA', ObB', OcC' concur at X = X(15258), a point with rather complicated coordinates which lies on K851 and on the lines X(4)X(6), X(20)X(648), X(95)X(253), etc.

K851 also contains the isogonal conjugates of the points U', V', W' described in the page K810.

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K851 belongs to the pencils generated by :

K003 and K187. See Table 54, line Q = X3 in the table. All these cubics pass through the foci of the inellipse with center X(3).

K002 and the union of the line at infinity and the Jerabek hyperbola. See Table 54, column P = X2 in the table. All these cubics pass through X(6) and the infinite points of K002.

K006 and K080. See Table 54, column P = S' in the table. All these cubics pass through the isogonal conjugates of the X3-OAP points. These points lie on the polar conic of X(3) in K851. This conic is the rectangular hyperbola (H) passing through X(3), X(20), X(54), X(155), X(2574), X(2575). The common tangent at X(3) to (H) and K851 is the Brocard axis.