Related papers: N-sociable Heronian Triangles
A Heron triangle is a triangle whose side lengths and area are integers. Two Heron triangles are amicable if the perimeter of one is the area of the other. We show, using elementary techniques, that there is only one pair of amicable Heron…
A convex polygon is Heronian if its side lengths and its area are integers. Two polygons are amicable if the area of one is equal to the perimeter of the other, and vice versa. We show that there are infinitely many pairs of amicable…
A Heron triangle is one that has all integer side lengths and integer area, which takes its name from Heron of Alexandria's area formula. From a more relaxed point of view, if rescaling is allowed, then one can define a Heron triangle to be…
A Heron triangle is a triangle whose side lengths and area are all positive integers. If the greatest common divisor of the three side lengths is $1$, it is called a primitive Heron triangle. In this paper, we give an equivalent condition…
We describe several algorithms for the generation of integer Heronian triangles with diameter at most $n$. Two of them have running time $\mathcal{O}\left(n^{2+\varepsilon}\right)$. We enumerate all integer Heronian triangles for $n\le…
Triangles with integer length sides and integer area are known as Heron triangles. Taking rescaling freedom into account, one can apply the same name when all sides and the area are rational numbers. A perfect triangle is a Heron triangle…
The relevance of this paper lies in the fact that it resolves two previously unsolved open problems. In the first part of the paper, a new lemma is proved, from which it follows that if there exists a triangle with integer sides and…
Two polygons are amicable if the perimeter of one is equal to the area of the other and vice versa. A polygon is a lattice polygon if its vertices are on the integer lattice $\Z^2$. We show that there is one pair of amicable lattice…
Each triangle has three exterior or external circles tangential to the three straight lines containing the three sides of the triangle.Among the preliminaries in this paper, is deriving formulas for the radii of the three exterior circles…
We define spherical Heron triangles (spherical triangles with "rational" side-lengths and angles) and parametrize them via rational points of certain families of elliptic curves. We show that the congruent number problem has infinitely many…
A Heron quadrilateral is a cyclic quadrilateral whose area and side lengths are rational. In this work, we establish a correspondence between Heron quadrilaterals and a family of elliptic curves of the form $y^2 = x3+/alpha x^2-n^2x.$ This…
Given any positive integer $n$, it is well-known that there always exists a triangle with rational sides $a,b$ and $c$ such that the area of the triangle is $n$. For a given prime $p \not \equiv 1$ modulo $8$ such that $p^{2}+1=2q$ for a…
A primitive Heron triangle is a triangle with integral sides and integral area where the greatest common divisor of the lengths of its sides is $1$. By utilizing the theory of elliptic curves, we prove that there exist infinitely many…
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…
In the early part of the paper, various geometrical formulas are derived. Then, at some point in the paper, the concept of a Pythagorean rational is introduced. A Pythagorean rational is a rational number which is the ratio of two integers…
A rational spherical triangle is a triangle on the unit sphere such that the lengths of its three sides and its area are rational multiples of $\pi$. Little and Coxeter have given examples of rational spherical triangles in 1980s. In this…
A rational triangle is a triangle with sides of rational lengths. In this short note, we prove that there exists a unique pair of a rational right triangle and a rational isosceles triangle which have the same perimeter and the same area.…
A polygon is equable if its area is equal to its perimeter. A pair of polygons is an amicable pair if the area of the first is equal to the perimeter of the second, and vice versa. A polygon is a lattice polygon if its vertices lie on the…
In this work, we define a triangle area number to be the area number of a triangle whose sides have integer lengths, and whose area is a rational number. In Result 3, on page 17, we prove that every triangle area number is in fact an…
In this paper, we study the function $H(a,b)$, which associates to every pair of positive integers $a$ and $b$ the number of positive integers $c$ such that the triangle of sides $a,b$ and $c$ is Heron, i.e., has integral area. In…