Maximum number of almost similar triangles in the plane

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.

On the Maximum Number of Dominating Classes in Graph Coloring

In this paper we investigate the dominating- -color number،  of a graph G. That is the maximum number of color classes that are also dominating when G is colored using colors. We show that where is the join of G and H. This result allows us to construct classes of graphs such that and thus provide some information regarding two questions raised in [1] and [2].  

The Maximum Number of Edges in a Strongly Multiplicative Graph

The Number of Unit-Area Triangles in the Plane: Theme and Variations∗

We show that the number of unit-area triangles determined by a set S of n points in the plane is O(n), improving the earlier bound O(n) of Apfelbaum and Sharir [2]. We also consider two special cases of this problem: (i) We show, using a somewhat subtle construction, that if S consists of points on three lines, the number of unit-area triangles that S spans, with one vertex on each of the lines...

Fuzzy plane geometry: triangles

A fuzzy triangle T (with a discrete-valued membership function) can be regarded as a nest of parallel-sided triangles T i with successively higher membership values. Such a nest is determined by its max projections on any two of its \sides". The area (perimeter) of T is a weighted sum of the areas (perimeters) of the T i 's. The side lengths and altitudes of T can also be deened as weighted sum...