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too complicated to be written here. Click on the link to download a text file. 

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 (JeanPierre 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. JeanPierre 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^62a^6+(a^4b^2c^2)(b^2+c^2)] / (b^2c^2) : : , not mentioned in the current edition of ETC. 
