[alogo] De Longchamps conic

Consider a point P and all lines PQ through that point. Take the orthopoles OPQ of these lines with respect to triangle ABC.
[1] The orthopoles OPQ generate an ellipse (known as orthopolar conic of P).
[2] There is a unique point for which this conic is tangent to all sides of the triangle. This is the De Longchamps point X(20) of the triangle (orthocenter of the anticomplementary of ABC).

[0_0] [0_1]
[1_0] [1_1]

It is well known (see ref. [5]) that the ellipse defined in [1] is tangent to the deltoid associated to triangle ABC, as the envelope of all its Wallace-Simson lines. The proof of [2] follows immediately from the fact that the simmultaneous tangency to the deltoid and the sides of the triangle can occur only at the points of tangency of the deltoid with the triangle which are the traces of X(69). Last point is the isotomic conjugate of the orthocenter H of ABC (denoted also by X(4)). Its traces {A'',B'',C''} are symmetric with respect to the middles of the sides to the feet of the altitudes {A',B',C'} of ABC (see Deltoid.html ).
This property implies that P must be on the intersection of the orthogonals to the sides at {A'',B'',C''} which identifies P with X(20) (it is symmetric to H with respect to the circumcenter).
Remark X(20) is on the Rigby pedal cubic defined as the locus of points P for which the projections {A'',B'',C''} joined to opposite vertices {A,B,C} define three concurrent lines.

[alogo] References

[1] Goormaghtigh, R. On Some Loci Connected with the Orthopole-GeometryThe American Mathematical Monthly, Vol. 37, No. 7. (Aug. - Sep., 1930), pp. 370-371.
[2] Kimberling, Clark Central Points and Central Lines in the Plane of a Triangle Mathematics Magazine 67 (1994)
[3] Kimberling, Clark. Encyclopedia of Triangle Centers http://faculty.evansville.edu/ck6/encyclopedia/ETC.html
[4] Pinkernell, M. Guido Cubic curves in the triangle plane Journal of Geometry 55(1966), pp. 141-161
[5] Ramler, J. O. The Orthopole Loci of Some One-Parameter Systems of Lines Referred to a Fixed Triangle The American Mathematical Monthly, Vol. 37, No. 3, (Mar., 1930), pp. 130-136

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