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6
votes
2answers
154 views

Intersecting Sets of Pythagorean Triples with Common Hypotenuses

For any $r\in\mathbb{N}$, let $A_r$ denote the set of all natural numbers that are potentially a side of a Pythagorean triple with hypotenuse $r$. Given any $N\in\mathbb{N}$, does there exist $r,s$ ...
3
votes
1answer
126 views

About the 'minimum triangle' which includes a convex bounded closed set

Question : Is the following true? "Letting $K$ be a convex bounded closed set on a plane, then there exists a triangle $M$, which includes $K$, such that $|M|\le 2|K|$. Here, $|M|,|K|$ is the area of ...
5
votes
3answers
386 views

Relationship between triangle free graphs and their minimum degree

Let $T$ be a triangle-free graph on $n$ vertices with minimum degree $\delta$ (which can be $0$). How does one show that $n >2\delta -1$? It seems to be true for bipartite graphs, but I cannot see ...
3
votes
2answers
516 views

Triangle area on surfaces of constant curvature

I am looking for an elementary derivation of the formula for the area of a geodesic triangle lying in a surface of constant curvature $\kappa$, depending on the angles and side length. Of course, the ...
4
votes
2answers
399 views

Are there Heronian triangles that can be decomposed into three smaller ones?

Is there anything known about the existence of Heronian triangles ABC (i.e. with rational side lengths and rational area) that can be decomposed into three Heronian triangles ABD, BCD, CAD? ...
2
votes
0answers
124 views

Series for envelope of triangle area bisectors

The lines which bisect the area of a triangle form an envelope as shown in this picture It is not difficult to show that the ratio of the area of the red deltoid to the area of the triangle is ...
10
votes
1answer
568 views

A generalization of the triangle counting problem for simple weighted graphs

One nice identity is $$tr(A^3)/6$$ which counts the number of triangles of a graph represented with adjacency matrix $A.$ It also implies that triangle counting can be performed in subcubic time. ...
1
vote
2answers
294 views

Triangles with Congruent Corresponding Sides that Cannot fold into a Tetrahedron

I've been trying to find, without much success, 4 triangles whose corresponding sides are congruent that cannot be folded into a tetrahedron. Anyone has any clue how to approach this problem?
5
votes
5answers
941 views

Impossible Heronian Triangles (Ratio of 2 Sides)

There is no Heronian triangle (or simply consider triangles on an integer lattice which also have integer side lengths) for which one side is half the length of another side. What other "side-side ...
9
votes
1answer
854 views

4-regular graphs with every edge in a triangle

I am interested in regular graphs in which every edge lies in a triangle. For 3-regular graphs, only the complete graph $K_4$ has this property, so there's not much to see here. For 4-regular ...