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X(2), X(4), X(6), X(13), X(14), X(15), X(16), X(30), X(524), X(1503) X(43273) → X(43277), X(43291), these points mentioned below 

Geometric properties : 

K1223 is the KHOcubic obtained when P = X(1503). See K1191 for explanations, also CL075 and the analogous cubic K1230. Its KHOequation is : √3 x y (y  2 z) + cotω [x^2 (y  z)  z (2 y  z) (y + z)] = 0 where ω is the Brocard angle. Note that [ ... ] = 0 is the cubic K1207. K1223 has three real asymptotes : (L1) = {30, 115, 187, 230}, (L2) = {524, 620, 2030}, (L3) = {1503, 2030}. (L1) and (L3) meet at X on the cubic and X is the common tangential of X(2), X(6), X(30), X(1503). X = X(43291) lies on the lines {2, 2418}, {4, 1384}, {5, 6}, {30, 115}, {32, 546}, {39, 3055}, {111, 468}, {140, 574}, etc. (L2) meets the cubic at Y, the common tangential of X(4), X(524) and X. The pivot X(1503) is the common tangential of X(13), X(14), X(15), X(16) hence the tangents to the cubic at these four points are parallel to the line X(4)X(6) and obviously to (L3) .

Other points on K1223 : • Q1 = {4,16} /\ {13,524} • Q2 = {4,15} /\ {14,524} • Q3 = {4,14} /\ {15,524} • Q4 = {4,13} /\ {16,524} Note that X lies on the lines {13,Q3}, {14,Q4},{15,Q1},{16,Q2} • Q5 = {2,1503} /\ {6,30} • Q0 = {4,524} /\ {X,Q5} 
