Forum Geometricorum, 1 (2001) 51 -- 58.
Abstract: Given a triangle, the isogonal conjugates of the infinite points of the side lines of the Morley (equilateral) triangle is an equilateral triangle PQR inscribed in the circumcircle. Their isotomic conjugates form another equilateral triangle P'Q'R' inscribed in the Steiner circum-ellipse, homothetic to PQR at the Steiner point. We show that under the one-to-one correspondence P --> P' between the circumcircle and the Steiner circum-ellipse established by isogonal and then isotomic conjugations, this is the only case when both PQR and P'Q'R' are equilateral.
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