Optimal Embedded and Enclosing Isosceles Triangles.

TRIANGLES; ANGLES; CONTAINERS

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. [ABSTRACT FROM AUTHOR]

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