Heron's Formula, Descartes Circles, and Pythagorean Triangles

This article reports the occurrence of binary quadratic forms in primitive Pythagorean triangles and their geometric interpretation. In addition to the well-known fact that the hypotenuse, z, of a right triangle, with sides of integral (relatively prime) length, can be expressed as the sum of two squares, z=a^2+b^2, where a and b are positive integers of opposite parity such that a>b>0 and gcd(a,b)=1, it is shown that the sum of the two sides, x and y, can also be expressed a...

A circle of curvature $n\in\mathbb{Z}^+$ is a part of finitely many primitive integral Apollonian circle packings. Each such packing has a circle of minimal curvature $-c\leq 0$, and we study the distribution of $c/n$ across all primitive integral packings containing a circle of curvature $n$. As $n\rightarrow\infty$, the distribution is shown to tend towards a picture we name the Apollonian staircase. A consequence of the staircase is that if we choose a random circle packin...

In this note we present a survey on some classical and modern approaches on Pythagorean triples. Some questions are also posed in direction of some materials under review. In particular some non commutative and operator theoretical approaches of Pythagorean triples are discussed

We define hyperbolic Heron triangles (hyperbolic triangles with "rational" side-lengths and area) and parametrize them in two ways as rational points of certain elliptic curves. We show that there are infinitely many hyperbolic Heron triangles with one angle $\alpha$ and area $A$ for any (admissible) choice of $\alpha$ and $A$; in particular, the congruent number problem has always infinitely many solutions in the hyperbolic setting. We also explore the question of hyperbolic...

Given an equilateral triangle with $a$ the square of its side length and a point in its plane with $b$, $c$, $d$ the squares of the distances from the point to the vertices of the triangle, it can be computed that $a$, $b$, $c$, $d$ satisfy $3(a^2+b^2+c^2+d^2)=(a+b+c+d)^2$. This paper derives properties of quadruples of nonnegative integers $(a,\, b,\, c,\, d)$, called triangle quadruples, satisfying this equation. It is easy to verify that the operation generating $(a,\, b,\...

It is shown that any primitive integral Apollonian circle packing captures a fraction of the prime numbers. Basically the method consists in applying the circle method, considering the curvatures produced by a well-chosen family of binary quadratic forms.

It is well known that Heron's theorem provides an explicit formula for the area of a triangle, as a symmetric function of the lengths of its sides. It has been extended by Brahmagupta to quadrilaterals inscribed in a circle (cyclic quadrilaterals). A natural problem is trying to further generalize the result to cyclic polygons with a larger number of edges, which, surprisingly, has revealed to be far from simple. In this paper we investigate such a problem by following a new ...

A formula for the radii and positions of four circles in the plane for an arbitrary linearly independent circle configuration is found. Among special cases is the recent extended Descartes Theorem on the Descartes configuration and an analytic solution to the Apollonian problem. The general theorem for n-spheres is also considered.

A geometric inequality among three triangles, originating in circle packing problems, is introduced. In order to prove it, we reduce the original formulation to the nonnegativity of a particular polynomial in four real indeterminates. Techniques based on sum of squares decompositions, semidefinite programming, and symmetry reduction are then applied to provide an easily verifiable nonnegativity certificate.

Consider two circles, externally tangential,and with integer radii R1, R2; and with R1>R2.The two circles have three tangent lines in common, one of them being T1T2. If M is the midpoint of T1T2, and K the point of intersection of the lines C1C2 and T1T2;then 16 right triangles are formed(C1 and C2 are the two circle centers), see Figure 1.In Section 6 of this paper, we find the precice form the two integers R1 and R2 must have, in order that the sixteen aforementioned right ...