Browsing by Author "Minevich, Igor"

Now showing 1 - 3 of 3
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    Synthetic foundations of cevian geometry, I: Fixed points of a ne maps in triangle geometry
    (Springer, 2016) Minevich, Igor; Morton, Patrick; Department of Mathematical Sciences, School of Science
    We give synthetic proofs of new results in triangle geometry, focusing especially on fixed points of certain affine maps which are defined in terms of the cevian triangles of a point P and its isotomic conjugate P′, with respect to a given triangle ABC. We give a synthetic proof of Grinberg’s formula for the cyclocevian map in terms of the isotomic and isogonal maps, and show that the complement Q of the isotomic conjugate P′ has many interesting properties. If TP is the affine map taking ABC to the cevian triangle DEF for P, it is shown that Q is the unique ordinary fixed point of TP when P does not lie on the sides of triangle ABC, its anticomplementary triangle, or the Steiner circumellipse of ABC. This paper forms the foundation for several more papers to follow, in which the conic on the 5 points A, B, C, P, Q is studied and its center is characterized as a fixed point of the map λ=TP′∘T−1Pλ=TP′∘TP−1.
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    Synthetic foundations of cevian geometry, III: the generalized orthocenter
    (Springer, 2016) Minevich, Igor; Morton, Patrick; Department of Mathematical Sciences, School of Science
    In this paper, the third in the series, we study the properties of the generalized orthocenter H corresponding to a point P, defined to be the unique point for which the lines HA, HB, HC are parallel, respectively, to QD, QE, QF, where DEF is the cevian triangle of P and Q=K∘ι(P)Q=K∘ι(P) is the isotomcomplement of P, both with respect to a given triangle ABC. We characterize the center Z of the cevian conic CPCP on the 5 points ABCPQ as the center of the affine map ΦP=TP∘K−1∘TP′∘K−1ΦP=TP∘K−1∘TP′∘K−1, where TP is the unique affine map for which TP(ABC) = DEF; TP' is defined similarly for the isotomic conjugate P′=ι(P)P′=ι(P) of P; and K is the complement map. The point Z is the point where the nine-point conic NHNH for the quadrangle ABCH and the inconic II of ABC, tangent to the sides at D, E, F, touch. This theorem generalizes the classical Feuerbach theorem.
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    Vertex Positions of the Generalized Orthocenter and a Related Elliptic Curve
    (2017) Minevich, Igor; Morton, Patrick; Mathematical Sciences, School of Science
    We study triangles $ABC$ and points $P$ for which the generalized orthocenter $H$ corresponding to $P$ coincides with a vertex $A,B$, or $C$. The set of all such points $P$ is a union of three ellipses minus $6$ points. In addition, if $T_P$ is the affine map taking $ABC$ to the cevian triangle $DEF$ of $P$ with respect to $ABC$, $P'$ is the isotomic conjugate of $P$, and $T_{P'}$ is the affine map taking $ABC$ to the cevian triangle of $P'$, then we study the locus of points $P$ for which the map $\textsf{M}_P=T_p \circ K^{-1} \circ T_{P'}$ is a translation. Here, $K$ is the complement map for $ABC$, and $\textsf{M}_P$ is an affine map taking the circumconic of $ABC$ for $P$ to the inconic of $ABC$ for $P$. The locus in question turns out to be an elliptic curve minus $6$ points, which can be synthetically constructed using the geometry of the triangle.