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In this paper, we extend the work of \cite{Chahal} in several directions. We first determine all Heron triangles that tightly circumscribe the unit circle and the associated $\tau$-congruent numbers generated by them. We then characterize…
We introduce and study arithmetic polygons. We show that these arithmetic polygons are connected to triples of square pyramidal numbers. For every odd $N\geq3$, we prove that there is at least one arithmetic polygon with $N$ sides. We also…
We introduce and study a new kind of congruent number problem on the right trapezoid.
Given any positive integer n, we prove the existence of infinitely many right triangles with area n and side lengths in certain number fields. This generalizes the famous congruent number problem. The proof allows the explicit construction…
In this paper we first study the isoperimetric problem in the case of integer triangles, as well as Alcuin's sequence and how it relates to the number of different integer triangles with a given perimeter. We then present and compare two…
Triangular numbers that are multiple of other triangular numbers are investigated. It is known that for any positive non-square integer multiplier, there is an infinity of multiples of triangular numbers which are triangular numbers. If the…
The conditions determining that two triangles are congruent play a basic role in planimetry. By comparing not congruent triangles with respect to given sets of corresponding elements it is important to discover if they have any common…
A tripartite-circle drawing of a tripartite graph is a drawing in the plane, where each part of a vertex partition is placed on one of three disjoint circles, and the edges do not cross the circles. The tripartite-circle crossing number of…
An N-tiling of triangle ABC by triangle T is a way of writing ABC as a union of N triangles congruent to T, overlapping only at their boundaries. The triangle T is the "tile". The tile may or may not be similar to ABC. We wish to understand…
It is known that, for any positive non-square integer multiplier $k$, there is an infinity of multiples of triangular numbers which are triangular numbers. We analyze the congruence properties of the indices $\xi$ of triangular numbers that…
Yet another example where "physical" (i.e. only checking finitely many special cases) gives a fully rigorous proof, notwithstanding what your "Intro To Proofs" prof told you!
We develop the basic and new tools for classifying non-side-to-side tilings of the sphere by congruent triangles. Then we prove that, if the triangle has any irrational angle in degree, such tilings are: a sequence of 1-parameter families…
We obtain new upper and lower bounds on the number of unit perimeter triangles spanned by points in the plane. We also establish improved bounds in the special case where the point set is a section of the integer grid.
We consider integers whose squares have just three decimal digits. Examples are e.g. given by $2108436491907081488939581538^2 = 4445504440405440505004450045555054500055550554550445444$ and $10100000000010401000000000101^2 =…
A research for a general expression of the triplets of integer sided triangles with a $120^{\circ}$ angle, in parallel with the case of a $60^{\circ}$ angle.
We study triangle decompositions of graphs. We consider constructions of classes of graphs where every edge lies on a triangle and the addition of the minimum number of multiple edges between already adjacent vertices results in a strongly…
We consider the problem of finding integer-sided triangles with R/r an integer, where R and r are the radii of the circumcircle and incircle respectively. We show that such triangles are relatively rare.
Considering regular graphs with every edge in a triangle we prove lower bounds for the number of triangles in such graphs. For r-regular graphs with r <= 5 we exhibit families of graphs with exactly that number of triangles and then…
For all positive non-square integer multiplier k, there is an infinity of multiples of triangular numbers which are also triangular numbers. With a simple change of variables, these triangular numbers can be found using solutions of Pell…
We give upper and lower bounds on the number of graphs of fixed degree which have a positive density of triangles. In particular, we show that there are very few such graphs, when compared to the number of graphs without this restriction.…