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When four tangents are drawn from a point M on the Thomson cubic to the Thomson cubic itself, their anharmonic ratio is constant (Salmon) and, up to a permutation of these tangents, is equal to : |
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which is obtained when the tangents are those drawn from K to the cubic namely AK, BK, CK, GK in this order. Now, given another pivotal cubic pK with pole Ω=p:q:r and pivot P=u:v:w, the tangents at A, B, C, P concur at P* and their anharmonic ratio is evaluated similarly. pK is said to be equivalent to the Thomson cubic if and only if these ratios are equal. This gives the condition : |
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For a given pivot P, the pole Ω must lie on the circum-conic with perspector X512 x P^2. For example, any pK with pivot G must have its pole on the circum-conic through G and K. For a given pole Ω, the pivot P must lie on the diagonal conic passing through the square roots of Ω and whose center is Ω÷X512. For example, any isogonal pK must have its pivot on the Steiner hyperbola i.e. the diagonal conic through the in/excenters and G. This is the polar conic of G in the Thomson cubic. Any projection or linear transformation, any isoconjugation with pole Q transform the Thomson cubic into an equivalent cubic. For example, the barycentric product of the Thomson cubic by a point Q is the cubic pK(X6 x Q^2, Q), a cubic equivalent to the Thomson cubic for any point Q. Any such cubic is called a multiple of the Thomson cubic (yellow lines in the table). The following table gives a selection of such cubics equivalent to the Thomson cubic. See also Table 68. |
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