Books Versus triangles

Similar Triangles, Another Trace of the Golden Ratio

In this paper we investigate similar triangles which are not congruent but have two pair congruent sides. We show that greatest lower bound of values for similarity ratio of such triangles is golden ratio. For right triangles, we prove that the supremum of values for similarity ratio is the square root of the golden ratio.

The Ill-Balanced Triangles of the Islamic States in Crisis

The Law of Tangents

Most geometry and pre-calculus books have entire units devoted to learning about two of the most important formulas in mathematics: the law of sines and the law of cosines. Both of these are very useful and very powerful tools, as they can be used to find unknown angles and sides of triangles. Yet, these books rarely devote anything to another formula that is equally powerful and useful: the la...

On Tensor Product of Graphs, Girth and Triangles

The purpose of this paper is to obtain a necessary and sufficient condition for the tensor product of two or more graphs to be connected, bipartite or eulerian. Also, we present a characterization of the duplicate graph $G 1 K_2$ to be unicyclic. Finally, the girth and the formula for computing the number of triangles in the tensor product of graphs are worked out.

Degree Relations of Triangles in Real-world Networks and Models

Triangles are an important building block and distinguishing feature of real-world networks, but their structure is still poorly understood. Despite numerous reports on the abundance of triangles, there is very little information on what these triangles look like. We initiate the study of degreelabeled triangles — specifically, degree homogeneity versus heterogeneity in triangles. This yields n...