Primitive Heronian Triangles With Integer Inradius and Exradii
نویسندگان
چکیده
It is well known that primitive Pythagorean triangles have integer inradius and exradii. We investigate the generalization to primitive Heronian triangles. In particular, we study the special cases of isosceles triangles and triangles with sides in arithmetic progression. We also give two families of primitive Heronian triangles, one decomposable and one indecomposable, which have integer inradii and exradii. When realized as lattice triangles, these two families have incenters and excenters at lattice points as well. Finally we pose two problems for further research.
منابع مشابه
Heronian Triangles Whose Areas Are Integer Multiples of Their Perimeters
We present an improved algorithm for finding all solutions to Goehl’s problem A = mP for triangles, i.e., the problem of finding all Heronian triangles whose area (A) is an integer multiple (m) of the perimeter (P ). The new algorithm does not involve elimination of extraneous rational triangles, and is a true extension of Goehl’s original method.
متن کاملHeronian Tetrahedra Are Lattice Tetrahedra
Extending a similar result about triangles, we show that each Heronian tetrahedron may be positioned with integer coordinates. More generally, we show the following: if an integral distance set in R can be positioned with rational coordinates, then it can in fact be positioned with integer coordinates. The proof, which uses the arithmetic of quaternions, is tantamount to an algorithm.
متن کاملAn Extension of the Fundamental Theorem on Right-angled Triangles
{3, 4, 5} is perhaps the most famous Pythagorean Triple with interest in such triples dating back many thousands of years to the ancient people of Mesopotamia. In this article, we shall consider such triples, with the restriction that the elements of these triples must not have any common factors they are Primitive Pythagorean Triples (PPTs). In particular, we shall consider the question of how...
متن کاملThe Delaunay Triangulation Maximizes the Mean Inradius
I prove that amongst all triangulations of a planar point set the Delaunay triangulation maximizes the arithmetic mean of the inradii of the triangles.
متن کاملMultiple attribute group decision making methods based on some normal neutrosophic number Heronian Mean operators
Abstract: In this paper, similar to the extension from intuitionistic fuzzy numbers (IFNs) to neutrosophic numbers (NNs), we propose the normal neutrosophic numbers (NNNs) based on the normal intuitionistic fuzzy numbers (NIFNs) to handle the incompleteness, indeterminacy and inconsistency of the evaluation information. In addition, because Heronian mean (HM) operators can consider capture the ...
متن کامل