Igor Minevich and Patrick Morton, A cevian locus and the geometric construction of a special elliptic curve,
Forum Geometricorum, 17 (2017) 449--464.

Abstract. Given a triangle ABC, we determine the locus L of points P, for which the affine mapping M = T_P \circ K^{-1} \circ T_{P'} is a half-turn, where T_P(ABC) = DEF is the cevian triangle of P, T_{P'}(ABC) is the cevian triangle of the isotomic conjugate P' of P, and K is the complement map, with respect to ABC. This completes the determination of the points P for which the inconic I = M(\tilde \cC_O) of P, tangent to the sides of ABC at the points D, E, F, is congruent to the circumconic \tilde \cC_O of ABC whose center is O=T_{P'}^{-1} \circ K(Q), where Q = K(P'). We show that the locus L is an elliptic curve minus six points, whose j-invariant is j= 2^4 11^3/5^2, and use the cevian geometry of ABC and P to give a synthetic construction of this elliptic curve.

[ps file] [pdf]