Rational angle bisectors on the coordinate plane and solutions of Pell's equations
Takashi Hirotsu
TL;DR
This work tackles the rational angle-bisection problem by linking slopes of angle bisectors to a Pell-type equation, specifically solving $(a-c)^2(b^2+1)=(b-c)^2(a^2+1)$ over $\mathbb{Q}$. It develops a constructive framework rooted in algebraic number theory: parametrizations arise from negative Pell equations $x^2-dy^2=-1$, the fundamental unit $\eta$ of $\mathbb{Q}(\sqrt d)$, and the arithmetic of $\mathbb{Z}[\sqrt d]$, including primes in $S(d)$ and their minimal exponents $l_p$. The paper presents two main parametric forms for rational solutions: one from pairs solving $x^2-y^2=-1$ and another from a shared negative Pell structure, expressed via $\alpha,\beta \in \mathbb{Q}(\sqrt d)$ with $N(\alpha)=N(\beta)=-1$, together with explicit parity conditions. Additionally, it gives a detailed description of strictly primitive integral solutions in terms of prime-splitting in $\mathbb{Q}(\sqrt d)$ and displays the resulting impact on rational angle-bisector construction, enriching the connections between Pell equations and lattice geometry.
Abstract
On the coordinate plane, the slopes $a$ and $b$ of two straight lines and the slope $c$ of one of their angle bisectors satisfy the equation $(a-c)^2(b^2+1) = (b-c)^2(a^2+1).$ Recently, an explicit formula for nontrivial integral solutions of this equation with solutions of negative Pell's equations was discovered by the author. In this article, for a given square-free integer $d > 1$ and a given integer $z > 1,$ we describe every integral solution $(x,y)$ of $|x^2-dy^2| = z$ such that $x$ and $dy$ are coprime by using the fundamental unit of $\mathbb Q(\sqrt d)$ and elements of $\mathbb Z[\sqrt d]$ whose absolute value of norms are the smallest prime powers. We also describe every nontrivial rational solution of the above equation as one of its applications.