too complicated to be written here. Click on the link to download a text file. X(3), X(25), X(64), X(66), X(3146), X(6525), X(33581), X(33582), X(33583), X(33584), X(33585) vertices of the cevian triangle of X(253) vertices of the tangential triangle of the Thomson triangle other points below Geometric properties :
 K1111 is an example of circum-cubic passing through the vertices of the tangential triangle of the Thomson triangle. See K172 and K1109, also K1110, K1112 and Table 27, paragraph Lemoine point of T(P). K1111 passes through its pseudo-pole Z1 = X(32) x X(253), on the lines {X3,X64}, {X6,X1661}, {X20,X801}, {X25,X800}, {X31,X1410}, {X98,X459}, etc, SEARCH = -1.23745515960379. K1111 meets the line at infinity at the same points as pK(X6, X1370). K1111 meets the circumcircle again at the same points as pK(X6, Po) where Po is the intersection of the lines {X2,X1350}, {X3,X51}, {X4,X343}, {X6,X22}, {X23,X154}, {X25,X394}, {X30,X1899}, {X52,X1181}, {X64,X3146}, etc, SEARCH = -4.97357115725543. Other points on K1111 : Z2 = a^2 (3 a^6-a^4 b^2+a^2 b^4-3 b^6-a^4 c^2-2 a^2 b^2 c^2+3 b^4 c^2+a^2 c^4+3 b^2 c^4-3 c^6) : : , SEARCH = -1.86575491623951 Z3 = a^4 (a^4-2 a^2 b^2+b^4+2 a^2 c^2+2 b^2 c^2-3 c^4)^2 (a^4+2 a^2 b^2-3 b^4-2 a^2 c^2+2 b^2 c^2+c^4)^2 (a^8-2 a^6 b^2+2 a^2 b^6-b^8-2 a^6 c^2+10 a^4 b^2 c^2-6 a^2 b^4 c^2-2 b^6 c^2-6 a^2 b^2 c^4+6 b^4 c^4+2 a^2 c^6-2 b^2 c^6-c^8) : : , SEARCH = 14.6562286111651 Z4 = a^2 (a^2+b^2-c^2) (a^2-b^2+c^2) (a^4-2 a^2 b^2+b^4+2 a^2 c^2+2 b^2 c^2-3 c^4) (a^4+2 a^2 b^2-3 b^4-2 a^2 c^2+2 b^2 c^2+c^4) (a^6+a^4 b^2-a^2 b^4-b^6+a^4 c^2-2 a^2 b^2 c^2+b^4 c^2-a^2 c^4+b^2 c^4-c^6) : : , SEARCH = -0.837164628108222 Z5 = a^2 (a^4-2 a^2 b^2+b^4+2 a^2 c^2+2 b^2 c^2-3 c^4) (a^4+2 a^2 b^2-3 b^4-2 a^2 c^2+2 b^2 c^2+c^4) (3 a^4+2 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+2 a^2 c^2-6 b^2 c^2+3 c^4) : : , SEARCH = 0.129137578513859 These points Z1, Z2, Z3, Z4, Z5, Po are now X(33581), X(33582), X(33583), X(33584), X(33585), X(33586) in ETC (2019-07-18).