Minimum Area Isosceles Containers

Kiss, Gergely; Pach, János; Somlai, Gábor

doi:10.2197/ipsjjip.28.759

Cited by 2 publications

(4 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…As Figure 21 illustrates, in both subcases we can find a smaller isosceles triangle (green in Figure 21) by shrinking the original triangle P RS so that the modified isosceles triangle contains ABC and has smaller area and perimeter. (ii) The proof of this case is analogous to the proof of case (iii) for the perimeter and the proof of [12][Lemma 3.2] for the area.…”

Section: Proof Of Theorem 13mentioning

confidence: 87%

“…The case of minimum area isosceles containers has been recently studied by Kiss, Pach, and Somlai [12]: they described all isosceles containers of a given triangle ∆ for which the minimum is attained. Here, we complete the picture: we characterize the optimal solutions of the other three problems stated above.…”

Section: Optimal Covers From a Classmentioning

confidence: 99%

“…Let ∆ be a triangle in R 2 and (i) let ∆ ⊇ ∆ be a minimum area isosceles container of ∆. Then ∆ and ∆ have a side in common and at one endpoint of this side they also have the same angle [12]; (ii) let ∆ ⊆ ∆ be a maximum area embedded isosceles triangle in ∆. Then ∆ and ∆ have a side in common and at one endpoint of this side they also have the same angle; (iii) let ∆ ⊆ ∆ be a maximum perimeter embedded isosceles triangle in ∆.…”

Section: Optimal Covers From a Classmentioning

confidence: 99%

“…The case of the area was handled in [12]. Concerning the perimeter, it is easy to see that there is a special container whose perimeter is at most twice as large as the one of ABC.…”

mentioning

confidence: 99%

See 3 more Smart Citations

Optimal Embedded and Enclosing Isosceles Triangles

Ambrus,

Csikós,

Kiss

et al. 2023

Int. J. Found. Comput. Sci.

Self Cite

View full text Add to dashboard Cite

Given a triangle [Formula: see text], we study the problem of determining the smallest enclosing and largest embedded isosceles triangles of [Formula: see text] 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 [Formula: see text] is the smallest area isosceles triangle containing [Formula: see text], then [Formula: see text] and [Formula: see text] share a side and an angle. In the present paper, we prove that for any triangle [Formula: see text], every maximum area isosceles triangle embedded in [Formula: see text] and every maximum perimeter isosceles triangle embedded in [Formula: see text] shares a side and an angle with [Formula: see text]. Somewhat surprisingly, the case of minimum perimeter enclosing triangles is different: there are infinite families of triangles [Formula: see text] whose minimum perimeter isosceles containers do not share a side and an angle with [Formula: see text].

show abstract

Section: Proof Of Theorem 13mentioning

confidence: 87%

Section: Optimal Covers From a Classmentioning

confidence: 99%

Section: Optimal Covers From a Classmentioning

confidence: 99%

“…The case of the area was handled in [12]. Concerning the perimeter, it is easy to see that there is a special container whose perimeter is at most twice as large as the one of ABC.…”

mentioning

confidence: 99%

See 2 more Smart Citations

Optimal Embedded and Enclosing Isosceles Triangles

Ambrus,

Csikós,

Kiss

et al. 2023

Int. J. Found. Comput. Sci.

Self Cite

View full text Add to dashboard Cite

show abstract

Optimal embedded and enclosing isosceles triangles

Ambrus¹,

Csikós²,

Kiss³

et al. 2022

Preprint

Self Cite

View full text Add to dashboard Cite

Given a triangle ∆, we study the problem of determining the smallest enclosing and largest embedded isosceles triangles of ∆ with respect to area and perimeter. This problem was initially posed by Nandakumar [16,18] and was first studied by Kiss, Pach, and Somlai [12], who showed that if ∆ is the smallest area isosceles triangle containing ∆, then ∆ and ∆ share a side and an angle. In the present paper, we prove that for any triangle ∆, every maximum area isosceles triangle embedded in ∆ and every maximum perimeter isosceles triangle embedded in ∆ shares a side and an angle with ∆. Somewhat surprisingly, the case of minimum perimeter enclosing triangles is different: there are infinite families of triangles ∆ whose minimum perimeter isosceles containers do not share a side and an angle with ∆.

show abstract

Minimum Area Isosceles Containers

Cited by 2 publications

References 17 publications

Optimal Embedded and Enclosing Isosceles Triangles

Optimal Embedded and Enclosing Isosceles Triangles

Optimal embedded and enclosing isosceles triangles

Contact Info

Product

Resources

About