A Point in Many 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

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.

On edge-disjoint empty triangles of point sets

Let P be a set of points in the plane in general position. Any three points x, y, x 2 P determine a triangle (x, y, z) of the plane. We say that (x, y, z) is empty if its interior contains no element of P . In this paper we study the following problems: What is the size of the largest family of edge-disjoint triangles of a point set? How many triangulations of P are needed to cover all the empt...

Subdivision by bisectors is dense in the space of all triangles

Starting with any nondegenerate triangle we can use a well defined interior point of the triangle to subdivide it into six smaller triangles. We can repeat this process with each new triangle, and continue doing so over and over. We show that starting with any arbitrary triangle, the resulting set of triangles formed by this process contains triangles arbitrarily close (up to similarity) any gi...

A Remark on Prime Divisors of Lengths of Sides of Heron Triangles

A Heron triangle is a triangle having the property that the lengths of its sides as well as its area are positive integers. Let P be a fixed set of primes, let S denote the set of integers divisible only by primes in P. We prove that there are only finitely many Heron triangles whose sides a, b, c ∈ S and are reduced, that is gcd(a, b, c) = 1. If P contains only one prime ≡ 1 (mod 4) then all t...