Notation: initial triangle T with corners P,Q,R; side lengths a,b,c; angles A,B,C; centre point S = (B_1 P + B_2 Q + B_3 R)/(B_1+B_2+B_3), where homogeneous barycentric coordinates B are some function of T; 3 cevian points PS.QR = U,V,W; 6 subsidiary triangles SPU etc. The 6 subsidiary centres appear to lie on a conic, for (all) triangle centres lying at the (1) centroid, B = [1,1,1]; (2) incentre, B = [a,b,c] or [sin A, ... ]; (3) e-centres, B = [-a,b,c], etc --- reliant on vertex ordering!; (4) circumcentre, B = [a^2(b^2+c^2-a^2), ... ] or [sin 2A, ... ]; (5) orthocentre B = [1/(b^2+c^2-a^2), ... ] or [tan A, ... ]; (6) first isogenic point X13, B = [a sec(A - pi/6), ... ]; In general (non-isosceles) they fail to lie on a conic for (7) symmedian point B = [a^2,b^2,c^2 or 1 - cos 2A, ... ]; (8) second isogenic point X14, B = [a sec(A + pi/6), ... ]; (9) fixed weight with B general constant vector (correcting earlier claim). The conjectures are based on evaluating the 6x6 determinant of quadratic monomials in the (projective) coordinate components. Fermat points (6),(7) cause particular difficulty, with the determinant evaluated only approximately: for example WFL's {12,17,25}-sided test triangle required at least 70 decimal-digit precision to confirm case (6), and case (7) was rejected with determinant ~ -2.538 E-13 . Fred Lunnon