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X(4), X(6), X(13), X(14), X(15), X(16), X(549), X(6560), X(6561), X(42136), X(42137)

X(43292) → X(43323)

other points below

Geometric properties :

Every pivotal KHO-cubic K(P) with pivot P on the van Aubel line (through H and K) passes through X(2), X(4), X(6), X(13), X(14), X(15), X(16), X(30). See K1207 for further details. K1224 is the locus of the tangential of H.

K1224 is a KHO-cubic, see K1191 for explanations and also CL075.

Its KHO-equation is : x^2 (y - z) - (5 y - z) z^2 = 0 or, equivalently, (x - z) (x + z) (y - z) - 4 y z^2 = 0.

K1224 is a crunodal cubic with node H. The nodal tangents meet the Brocard axis at the KHO-points (±√5,0,1).

X(6) is a point of inflexion with tangent passing through X(5) and harmonic polar the Euler line. Hence, the tangent at X(549) passes through X(6).

K1224 meets the Kiepert hyperbola at H (twice), X(13), X(14) and two other imaginary points, on the line X(6)X(140), which are the KHO-points (±3i,1,3).

K1224 meets the Evans conic at X(13), X(14), X(15), X(16) and two other real points E5, E6 on the line X(6)X(550) which are the KHO-points (±3√2,1,-3).

Other KHO-points on the curve : (±25,12,-15), (±25,6,5), (±7,30,3), (±9,260,4).

Parametrization :

For any real (sometimes complex) number t or infinity (giving X549), the KHO-point P(t) = (5 t^2 - 1, t (t^2 - 1), t (5 t^2 - 1)) lies on K1224. This gives a lot of simple points on the curve. Note that X(6), P(t), P(-t) are collinear.

The third point of the cubic on the line passing through P(t1), P(t2) is P(t3) where t3 = - (t1 + t2) / (5 t1 t2 + 1).

Hence, the tangential of P(t) is P(T) where T = - 2 t / (5 t^2 + 1).

***

A generalization

The cubic (K) with the following KHO-equation is a cubic of the same type : x^2 (v y + w z) + z^2 (q y + r z) = 0, v ≠ 0.

H is a singular point and K is a point of inflexion with tangent v y + w z = 0 and harmonic polar the Euler line.

(K) meets the Euler and van Aubel lines again at the KHO-points (0,r,-q) and (w,-v,0) respectively.

The polar conic of H degenerates and has KHO-equation : v x^2 + q z^2 = 0. From this, it can be seen that (K) is

• a crunodal cubic if and only if vq < 0. See K1224, K1226 for example. In this case, (K) has only one real point of inflexion, namely K.

• an acnodal cubic if and only if vq > 0. See K1225, K1227, K1229 for example. In this case, (K) has three real points of inflexion, namely K and two others F1, F2, collinear on the line having KHO-equation : 4 q v y + (3 r v + q w) z = 0. The tangents to (K) at F1, F2 meet on the Euler line at (0,q w - 9 r v,8 q v).

• a cuspidal cubic if and only if q = 0. See K1228 for example. The cuspidal tangent is the Euler line.

The five mentioned cubics all pass through X(13), X(14), X(15), X(16) and, more generally, (K) passes through these same points if and only if r + w = 0 and q + 9 v + 4 w = 0. The polar conic of H becomes v x^2 - (9 v + 4 w) z^2 = 0.

This polar conic is the Euler line (counted twice) when 9 v + 4 w = 0 and then, (K) is precisely the cuspidal cubic K1228.

(K) meets the Kiepert hyperbola at H (twice), X(13), X(14) and two other points K5 and K6, on the KHO-line 3 v y + w z = 0. These points are real and distinct if and only if v (3 v + 2 w) < 0. They coincide when 3 v + 2 w = 0, corresponding to the cubic K1226, tangent at G to the Kiepert hyperbola.

(K) meets the Evans conic at X(13), X(14), X(15), X(16) and two other points E5 and E6, on the KHO-line 3 v y + (4 v + 3w) z = 0. These points are always real and distinct.