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K1223

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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 KHO-cubic obtained when P = X(1503). See K1191 for explanations, also CL075 and the analogous cubic K1230.

Its KHO-equation 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) .

K1223a

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}