too complicated to be written here. Click on the link to download a text file. X(6), X(325), X(567), X(1344), X(1345), X(3519), X(4846)
 A line (L) passing through the Lemoine point K meets the Euler line at Q and the Evans conic (EC) at Q1, Q2. The pole of (L) in (EC) is a point E on the Euler line hence the tangents at Q1, Q2 to (EC) pass through E. This point E is the reflection about G of the inverse of Q in the circle with center G, radius OG / √2. The trilinear pole T of (L) lies on the circumcircle and the line (D) passing through E and T meets (L) at X. When (L) rotates about K, the locus of X is Q166. Q166 has a triple point at K hence (L) meets Q166 again at one and only one other point unless (L) is one of the three tangents at K. The following table gives a selection of other points on Q166.
 Q 1st barycentric of X SEARCH X4 (a^2+b^2-c^2)^2 (a^2-b^2+c^2)^2 (a^6 b^2-3 a^4 b^4+3 a^2 b^6-b^8+a^6 c^2-8 a^4 b^2 c^2+5 a^2 b^4 c^2+2 b^6 c^2-3 a^4 c^4+5 a^2 b^2 c^4-2 b^4 c^4+3 a^2 c^6+2 b^2 c^6-c^8) -2.863601524897933 X5 (a^4-2 a^2 b^2+b^4-2 b^2 c^2+c^4) (a^4+b^4-2 a^2 c^2-2 b^2 c^2+c^4) (a^8-3 a^6 b^2+3 a^4 b^4-a^2 b^6-3 a^6 c^2+6 a^4 b^2 c^2-a^2 b^4 c^2-2 b^6 c^2+3 a^4 c^4-a^2 b^2 c^4+4 b^4 c^4-a^2 c^6-2 b^2 c^6) 1.237211930360331 X20 (a^4+6 a^2 b^2+b^4-2 a^2 c^2-2 b^2 c^2+c^4) (a^4-2 a^2 b^2+b^4+6 a^2 c^2-2 b^2 c^2+c^4) (16 a^8-39 a^6 b^2+21 a^4 b^4+11 a^2 b^6-9 b^8-39 a^6 c^2-12 a^4 b^2 c^2+65 a^2 b^4 c^2-14 b^6 c^2+21 a^4 c^4+65 a^2 b^2 c^4+46 b^4 c^4+11 a^2 c^6-14 b^2 c^6-9 c^8) 5.789798783580428 X140 (a^4-2 a^2 b^2+b^4-4 a^2 c^2-2 b^2 c^2+c^4) (a^4-4 a^2 b^2+b^4-2 a^2 c^2-2 b^2 c^2+c^4) (a^8-4 a^6 b^2+6 a^4 b^4-4 a^2 b^6+b^8-4 a^6 c^2+8 a^4 b^2 c^2-4 b^6 c^2+6 a^4 c^4+6 b^4 c^4-4 a^2 c^6-4 b^2 c^6+c^8) 15.4089653502394 X376 (a^4+10 a^2 b^2+b^4-2 a^2 c^2-2 b^2 c^2+c^4) (a^4-2 a^2 b^2+b^4+10 a^2 c^2-2 b^2 c^2+c^4) (16 a^8-43 a^6 b^2+33 a^4 b^4-a^2 b^6-5 b^8-43 a^6 c^2+16 a^4 b^2 c^2+49 a^2 b^4 c^2-22 b^6 c^2+33 a^4 c^4+49 a^2 b^2 c^4+54 b^4 c^4-a^2 c^6-22 b^2 c^6-5 c^8) 3.148498048097796 X381 (a^2-a b+b^2-c^2) (a^2+a b+b^2-c^2) (a^2-b^2-a c+c^2) (a^2-b^2+a c+c^2) (a^8-4 a^6 b^2+6 a^4 b^4-4 a^2 b^6+b^8-4 a^6 c^2+13 a^4 b^2 c^2-5 a^2 b^4 c^2-4 b^6 c^2+6 a^4 c^4-5 a^2 b^2 c^4+6 b^4 c^4-4 a^2 c^6-4 b^2 c^6+c^8) 0.1382366841135917 X382 (a^4+3 a^2 b^2+b^4-2 a^2 c^2-2 b^2 c^2+c^4) (a^4-2 a^2 b^2+b^4+3 a^2 c^2-2 b^2 c^2+c^4) (5 a^8-6 a^6 b^2-12 a^4 b^4+22 a^2 b^6-9 b^8-6 a^6 c^2-51 a^4 b^2 c^2+49 a^2 b^4 c^2+8 b^6 c^2-12 a^4 c^4+49 a^2 b^2 c^4+2 b^4 c^4+22 a^2 c^6+8 b^2 c^6-9 c^8) -19.57459438843124 X546 (3 a^4+4 a^2 b^2+3 b^4-6 a^2 c^2-6 b^2 c^2+3 c^4) (3 a^4-6 a^2 b^2+3 b^4+4 a^2 c^2-6 b^2 c^2+3 c^4) (5 a^8-24 a^6 b^2+42 a^4 b^4-32 a^2 b^6+9 b^8-24 a^6 c^2+96 a^4 b^2 c^2-44 a^2 b^4 c^2-28 b^6 c^2+42 a^4 c^4-44 a^2 b^2 c^4+38 b^4 c^4-32 a^2 c^6-28 b^2 c^6+9 c^8) -0.5687088983668553 X547 (5 a^4-4 a^2 b^2+5 b^4-10 a^2 c^2-10 b^2 c^2+5 c^4) (5 a^4-10 a^2 b^2+5 b^4-4 a^2 c^2-10 b^2 c^2+5 c^4) (7 a^8-16 a^6 b^2+6 a^4 b^4+8 a^2 b^6-5 b^8-16 a^6 c^2+16 a^4 b^2 c^2+4 a^2 b^4 c^2-4 b^6 c^2+6 a^4 c^4+4 a^2 b^2 c^4+18 b^4 c^4+8 a^2 c^6-4 b^2 c^6-5 c^8) -13.1837642724211 X548 (a^4+12 a^2 b^2+b^4-2 a^2 c^2-2 b^2 c^2+c^4) (a^4-2 a^2 b^2+b^4+12 a^2 c^2-2 b^2 c^2+c^4) (35 a^8-96 a^6 b^2+78 a^4 b^4-8 a^2 b^6-9 b^8-96 a^6 c^2+48 a^4 b^2 c^2+100 a^2 b^4 c^2-52 b^6 c^2+78 a^4 c^4+100 a^2 b^2 c^4+122 b^4 c^4-8 a^2 c^6-52 b^2 c^6-9 c^8) 2.635052872950796 X549 (a^4-2 a^2 b^2+b^4-8 a^2 c^2-2 b^2 c^2+c^4) (a^4-8 a^2 b^2+b^4-2 a^2 c^2-2 b^2 c^2+c^4) (5 a^8-17 a^6 b^2+21 a^4 b^4-11 a^2 b^6+2 b^8-17 a^6 c^2+26 a^4 b^2 c^2+5 a^2 b^4 c^2-14 b^6 c^2+21 a^4 c^4+5 a^2 b^2 c^4+24 b^4 c^4-11 a^2 c^6-14 b^2 c^6+2 c^8) -16.20229460099703 X550 (a^4+8 a^2 b^2+b^4-2 a^2 c^2-2 b^2 c^2+c^4) (a^4-2 a^2 b^2+b^4+8 a^2 c^2-2 b^2 c^2+c^4) (5 a^8-13 a^6 b^2+9 a^4 b^4+a^2 b^6-2 b^8-13 a^6 c^2+2 a^4 b^2 c^2+17 a^2 b^4 c^2-6 b^6 c^2+9 a^4 c^4+17 a^2 b^2 c^4+16 b^4 c^4+a^2 c^6-6 b^2 c^6-2 c^8) 3.986053788375043 X631 (a^4-2 a^2 b^2+b^4-6 a^2 c^2-2 b^2 c^2+c^4) (a^4-6 a^2 b^2+b^4-2 a^2 c^2-2 b^2 c^2+c^4) (16 a^8-57 a^6 b^2+75 a^4 b^4-43 a^2 b^6+9 b^8-57 a^6 c^2+96 a^4 b^2 c^2+11 a^2 b^4 c^2-50 b^6 c^2+75 a^4 c^4+11 a^2 b^2 c^4+82 b^4 c^4-43 a^2 c^6-50 b^2 c^6+9 c^8) -722.1374719210553 X632 (3 a^4-6 a^2 b^2+3 b^4-8 a^2 c^2-6 b^2 c^2+3 c^4) (3 a^4-8 a^2 b^2+3 b^4-6 a^2 c^2-6 b^2 c^2+3 c^4) (7 a^8-39 a^6 b^2+75 a^4 b^4-61 a^2 b^6+18 b^8-39 a^6 c^2+102 a^4 b^2 c^2-13 a^2 b^4 c^2-50 b^6 c^2+75 a^4 c^4-13 a^2 b^2 c^4+64 b^4 c^4-61 a^2 c^6-50 b^2 c^6+18 c^8) 8.871749178621794 X1316 (a^4+b^4-a^2 c^2-b^2 c^2) (a^4-a^2 b^2-b^2 c^2+c^4) (2 a^10-6 a^8 b^2+7 a^6 b^4-5 a^4 b^6+3 a^2 b^8-b^10-6 a^8 c^2+10 a^6 b^2 c^2-5 a^4 b^4 c^2-5 a^2 b^6 c^2+3 b^8 c^2+7 a^6 c^4-5 a^4 b^2 c^4+10 a^2 b^4 c^4-2 b^6 c^4-5 a^4 c^6-5 a^2 b^2 c^6-2 b^4 c^6+3 a^2 c^8+3 b^2 c^8-c^10) 43.22408929306134
 Note : Peter Moses recently introduced a lot of new centers on the Evans conic. See preamble just before X(41943) in ETC.