Minimum area isosceles containers

We show that every minimum area isosceles triangle containing a given triangle $T$ shares a side and an angle with $T$. This proves a conjecture of Nandakumar motivated by a computational problem. We use our result to deduce that for every triangle $T$, (1) there are at most $3$ minimum area isosceles triangles that contain $T$, and (2) there exists an isosceles triangle containing $T$ whose area is smaller than $\sqrt2$ times the area of $T$. Both bounds are best possible

Optimal embedded and enclosing isosceles triangles

Given a triangle Δ, we study the problem of determining the smallest enclosing and largest embedded isosceles triangles of Δ with respect to area and perimeter. This problem was initially posed by Nandakumar and was first studied by Kiss, Pach, and Somlai, who showed that if Δ′ is the smallest area isosceles triangle containing Δ, then Δ′ and Δ share a side and an angle. In the present paper, we prove that for any triangle Δ, every maximum area isosceles triangle embedded in Δ and every maximum perimeter isosceles triangle embedded in Δ shares a side and an angle with Δ. Somewhat surprisingly, the case of minimum perimeter enclosing triangles is different: there are infinite families of triangles Δ whose minimum perimeter isosceles containers do not share a side and an angle with Δ

Optimal embedded and enclosing isosceles triangles

Given a triangle Δ, we study the problem of determining the smallest enclosing and largest embedded isosceles triangles of Δ with respect to area and perimeter. This problem was initially posed by Nandakumar [17, 22] and was first studied by Kiss, Pach, and Somlai [13], who showed that if Δ′ is the smallest area isosceles triangle containing Δ, then Δ′ and Δ share a side and an angle. In the present paper, we prove that for any triangle Δ, every maximum area isosceles triangle embedded in Δ and every maximum perimeter isosceles triangle embedded in Δ shares a side and an angle with Δ. Somewhat surprisingly, the case of minimum perimeter enclosing triangles is different: there are infinite families of triangles Δ whose minimum perimeter isosceles containers do not share a side and an angle with Δ

Terrestrial radioisotopes in black shale hosted Mn-carbonate deposit (Úrkút, Hungary)

Previously, little attention has been paid to terrestrial radioisotopes (U, Th, 40K) occurring in manganese ores, despite the fact that the biogeochemical relationship between Mn and U is versatile. Occurrence of terrestrial radioisotopes in great amounts during mining on a long-term causes significant radiation exposure. It is important to inspect black shale-hosted manganese ores from this aspect, as black shales are typically potential U-rich formations. Despite the increased radon concentration in the mine, based on the detailed major elements, trace elements and gamma spectroscopy inspection of the rock types of deposit, the U, Th enrichment was undetectable. However, the U and Th content of about average terrestrial abundance of the great ore amount may be in the background of the increased radon concentration level. This Mn-carbonate ore deposit in spite of the low U content exhibit potential radon danger for miners, which can be eliminated with intensive air change only. © 2013 Versita Warsaw and Springer-Verlag Wien