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Let O1, and H1 be two points, in the plane of any triangle of reference ABC, so related that if 01P, O1Q, O1R be the perpendiculars drawn to the sides of ABC, then AP, BQ, CR meet in H1. We shall find that O1, and H1 describe respectively two cubics which are related to each other in a remarkable manner. We shall show, for instance, that points in each curve may be derived from each other by two sets of three alternative rational quadric transformations, and that the join of correspondents passes through a fixed point as in plane projection. We shall then discuss the homographic relation between corresponding pencils formed by rays through pairs of related points—not direct correspondents—and investigate the relation between these latter points.
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* This is, of course, the definition of the isotomio conjugate of a point. Also the isotomic conjugate of (α β γ) is 
† By Ceva's Theorem we may easily derive, for a point on the O-loous, that lmn=l′m′n′, which is equation (1).
* P 2′ is the isogonal conjugate of P 1′, and, according to our notation, should be written P′. Unfortunately, this clashes with the notation for opposites, since P 2′ and P are not opposites.
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