[1] The orthopoles O

[2] There is a unique point for which this conic is tangent to all sides of the triangle. This is the

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).

[2] Kimberling, Clark

[3] Kimberling, Clark.

[4] Pinkernell, M. Guido

[5] Ramler, J. O.

Produced with EucliDraw© |