arXiv subjects
Publications and source records attributed to Patrick Morton.
We give an affine proof of Feuerbach's theorem, by constructing an explicit affine map which takes the nine-point circle of any given Euclidean triangle to the incircle and fixes the Feuerbach point. The proof is shown to be valid in any Hilbert plane which satisfies the parallel postulate.
We study the elliptic curve $E_a: (ax+1)y^2+(ax+1)(x-1)y+x^2-x=0$, which we call the geometric normal form of an elliptic curve. We show that any elliptic curve whose $j$-invariant is real is isomorphic to a curve $E_a$ in geometric normal form, and show that for $a \notin \{0, -1, -9\}$, the points on $E_a$, minus a set of $6$ points, can be characterized in terms of the cevian geometry of a triangle.
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.
In this paper we give a completely synthetic proof of the TCC-perspector theorem, that the isogonal conjugate $γ(H)$ of the generalized orthocenter $H$ (defined in Part III of this series of papers), with respect to a triangle $ABC$ and a point $P$, is the perspector of the tangential triangle of $ABC$ and the circumcevian triangle (both with respect to the circumcircle) of the isogonal conjugate $γ(Q)$, where $Q$ is the complement of the isotomic conjugate $P'$ of the point $P$.
In a previous paper we defined the circumconic of a triangle $ABC$ with respect to a point $P$ as the conic $\tilde C=T_{P'}^{-1}(N_{P'})$, where $N_{P'}$ is the $9$-point conic for the quadrangle $ABCP'$ with respect to the line at infinity, $P'$ is the isotomic conjugate of $P$ with respect to $ABC$, and $T_{P'}$ is the affine map taking $ABC$ to the cevian triangle for $P'$. In this paper we determine the locus of points for which a certain affine map $\textsf{M}$ taking the circumconic $\tilde C$ to the inconic $\mathcal{I}$, defined to be the unique conic tangent to the sides of $ABC$ at the traces of the point $P$ on those sides, is a half-turn. This locus turns out to be an elliptic curve minus six points, which can be constructed geometrically using a family of affine maps defined for points on three open arcs of a circle.
Solutions of the quartic Fermat equation in ring class fields of odd conductor over quadratic fields $K=\mathbb{Q}(\sqrt{-d})$ with $-d \equiv 1$ (mod $8$) are shown to be periodic points of a fixed algebraic function $T(z)$ defined on the punctured disk $0< |z|_2 \le \frac{1}{2}$ of the maximal unramified, algebraic extension $\textsf{K}_2$ of the $2$-adic field $\mathbb{Q}_2$. All ring class fields of odd conductor over imaginary quadratic fields in which the prime $p=2$ splits are shown to be generated by complex periodic points of the algebraic function $T$, and conversely, all but two of the periodic points of $T$ generate ring class fields over suitable imaginary quadratic fields. This gives a dynamical proof of a class number relation originally proved by Deuring. It is conjectured that a similar situation holds for an arbitrary prime $p$ in place of $p=2$, where the case $p=3$ has been previously proved by the author, and the case $p=5$ will be handled in Part II.
This paper continues the investigation of Part I, by studying the conic $\mathcal{C}_P$ on the five points $ABCPQ$, where $ABC$ is a given ordinary triangle and $Q$ is the isotomcomplement of $P$, defined as the complement of the isotomic conjugate $P'$ of $P$ with respect to triangle $ABC$. We show that $\mathcal{C}_P$ also lies on the points $P'$ and $Q'$, where $Q'$ is the isotomcomplement of $P'$. The conic $\mathcal{C}_P$ lies on six other points which are the images of the vertices of $ABC$ under the affine mapping $λ=T_{P'} \circ T_P^{-1}$ and its inverse, where $T_P$ and $T_{P'}$ are the unique affine maps taking $ABC$ to the cevian triangles of $P$ and $P'$, respectively. In the paper we characterize the center $Z$ of $\mathcal{C}_P$ as the unique fixed point of $λ$ in the extended plane, when $\mathcal{C}_P$ is a parabola or an ellipse, and the unique ordinary fixed point of $λ$, when $\mathcal{C}_P$ is a hyperbola. We also show that $Z=GV \cdot T_P(GV)$, where $G$ is the centroid of $ABC$ and $V=PQ \cdot P'Q'$. When $P$ is the Gergonne point of $ABC$, this gives a new characterization of the Feuerbach point $Z$. All of our arguments are purely synthetic.
In this paper, the third in the series, we define the generalized orthocenter $H$ corresponding to a point $P$, with respect to triangle $ABC$, as 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 $Q=K \circ ι(P)$ is the $isotomcomplement$ of $P$, both with respect to $ABC$. We prove a generalized Feuerbach Theorem, and characterize the center $Z$ of the cevian conic $\mathcal{C}_P$, defined in Part II, as the center of the affine map $Φ_P = T_P \circ K^{-1} \circ T_{P'} \circ K^{-1}$, where $T_P$ is the unique affine map for which $T_P(ABC)=DEF$; $T_{P'}$ is defined similarly for the isotomic conjugate $P'=ι(P)$ of $P$; and $K$ is the complement map. The affine map $Φ_P$ fixes $Z$ and takes the nine-point conic $\mathcal{N}_H$ for the quadrangle $ABCH$ (with respect to the line at infinity) to the inconic $\mathcal{I}$, defined to be the unique conic which is tangent to the sides of $ABC$ at the points $D, E, F$. The point $Z$ is therefore the point where the nine-point conic $\mathcal{N}_H$ and the inconic $\mathcal{I}$ touch. This theorem generalizes the usual Feuerbach theorem and holds in all cases where the point $P$ is not on a median, whether the conics involved are ellipses, parabolas, or hyperbolas, and also holds when $Z$ is an infinite point. We also determine the locus of points $P$ for which the generalized orthocenter $H$ coincides with a vertex of $ABC$; this locus turns out to be the union of three conics minus six points. All our proofs are synthetic, and combine affine and projective arguments.
Explicit solutions of the cubic Fermat equation are constructed in ring class fields $Ω_f$, with conductor $f$ prime to $3$, of any imaginary quadratic field $K$ whose discriminant satisfies $d_K \equiv 1$ (mod $3$), in terms of the Dedekind $η$-function. As $K$ and $f$ vary, the set of coordinates of all solutions is shown to be the exact set of periodic points of a single algebraic function and its inverse defined on natural subsets of the maximal unramified, algebraic extension $\textsf{K}_3$ of the $3$-adic field $\mathbb{Q}_3$. This is used to give a dynamical proof of a class number relation of Deuring. These solutions are then used to give an unconditional proof of part of Aigner's conjecture: the cubic Fermat equation has a nontrivial solution in $K=\mathbb{Q}(\sqrt{-d})$ if $d_K \equiv 1$ (mod $3$) and the class number $h(K)$ is not divisible by $3$. If $3 \mid h(K)$, congruence conditions for the trace of specific elements of $Ω_f$ are exhibited which imply the existence of a point of infinite order in $Fer_3(K)$.
We give synthetic proofs of many new results in triangle geometry, focusing especially on fixed points of certain affine maps which are defined in terms of the cevian triangle $DEF$ of a point $P$ with respect to a given triangle $ABC$, as well as the cevian triangle of the isotomic conjugate $P'$ of $P$ with respect to $ABC$. We prove a formula for the cyclocevian map in terms of the isotomic and isogonal maps using an entirely synthetic argument, and show that the complement $Q$ of the isotomic conjugate $P'$ has many interesting properties. If $T_P$ is the affine map taking $ABC$ to $DEF$, we show synthetically that $Q$ is the unique ordinary fixed point of $T_P$ when $P$ is any point not lying on the sides of triangle $ABC$, its anti-complementary triangle, or the Steiner circumellipse of $ABC$. We also show that $T_P(Q')=P$ if $Q'$ is the complement of $P$, and that the affine map $T_P T_{P'}$ is either a homothety or a translation which always has the $P$-ceva conjugate of $Q$ as a fixed point. Finally, we show that $P$ lies on the Steiner circumellipse if and only if $T_PT_{P'}=K^{-1}$, where $K$ is the complement map for $ABC$. 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 $λ=T_{P'} T_P^{-1}$.
It is shown that the quartic Fermat equation $x^4 +y^4=1$ has nontrivial integral solutions in the Hilbert class field $Σ$ of any quadratic field $K=\mathbb{Q}(\sqrt{-d})$ whose discriminant satisfies $-d \equiv 1$ (mod 8). A corollary is that the quartic Fermat equation has no nontrivial solution in $K=\mathbb{Q}(\sqrt{-p})$, for $p$ $( > 7)$ a prime congruent to $7$ (mod 8), but does have a nontrivial solution in the odd degree extension $Σ$ of $K$. These solutions arise from explicit formulas for the points of order 4 on elliptic curves in Tate normal form. The solutions are studied in detail and the results are applied to prove several properties of the Weber singular moduli introduced by Yui and Zagier.