One property of the Darboux cubic
3 cevians AA1, BB1, CC1 intersect at D. draw 6 circles: (1) with center at A1 going through B (2) with center at A1 going through C (3) with center at B1 going through A (4) with center at B1 going through C (5) with center at C1 going through A (6) with center at C1 going through B Circles (1), (6) (2),(4) (3),(5) intersect at B', C' A' accordingly. Then AA', BB', CC' always intersect at X in these special cases: D X X(2) X(4) X(4) X(3) X(7) X(1) X(8) X(84) in this case O_A, O_B, O_C are collinear. X(20) X(3346) X(69) X(64) X(189) X(40) X(253) X(20) X(329) X(3345) X1032 X(1498) X1034 X(1490) K007 K004 interestingly all the points in the second column lie on the Darboux cubic a...