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X(2), X(6)

P1, P2, P1*, P2* defined below

The trilinear polar of P meets the sidelines of ABC at U, V, W. Ub, Uc are the projections of U on AC, AB and Vc, Va, Wa, Wb are defined similarly.

The algebric areas of triangles UbVcWa and UcVaWb are opposite if and only if P lies on the Thomson cubic (Jean-Pierre Ehrmann). They are equal if and only if P lies on K231.

K231 is an isogonal nK with root X(20), the de Longchamps point. It is a member of the class CL061.

It meets the Thomson cubic at A, B, C, G, K and four (not always real) other points P1, P2 and their isogonal conjugates P1*, P2*. For these four latter points, the points Ub, Vc, Wa and Uc, Va, Wb are collinear and the two corresponding lines are parallel.

Jean-Pierre Ehrmann found the following characterization for these four points. Draw from the de Longchamps point the lines tangent to the Steiner circumellipse; these lines touch the ellipse at S1, S2. P1, P1* - or P2, P2* - is the pair of isogonal conjugates on the trilinear polar of S1 - or S2 - (note that these trilinear polars intersect at G and go through the infinite points of the circumconic with perspector the reflection X(376) of G in O ).

The third point of K231 on GK is E = [b^6+c^6-2a^6+(a^4-b^2c^2)(b^2+c^2)] / (b^2-c^2) : : , not mentioned in the current edition of ETC.