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.

Abstract We say that a triangle T tiles polygon A , if can be dissected into finitely many nonoverlapping triangles similar to . show $$N&gt;42$$ N &gt; 42 then there are at most three nonsimilar such the angles of rational multiples $$\pi $$ π<...

A triangle $T'$ is $\varepsilon$-similar to another $T$ if their angles pairwise differ by at most $\varepsilon$. Given a $T$, $\varepsilon>0$ and $n\in\mathbb{N}$, B\'ar\'any F\"uredi asked determine the maximum number of triangles $h(n,T,\varepsilon)$ being in planar point set size $n$. We show that for almost all there exists $\varepsilon=\varepsilon(T)>0$ such $h(n,T,\varepsilon)=n^3/24 (1+...