Editing Square (section)

===Inscribed squares===
[[File:Calabi triangle.svg|thumb|The [[Calabi triangle]] and the three placements of its largest square.{{sfnp|Conway|Guy|1996|p=[https://books.google.com/books?id=0--3rcO7dMYC&pg=PA206 206]}} The placement on the long side of the triangle is inscribed; the other two are not.]]
{{Main|Inscribed square problem|Inscribed square in a triangle}}
A square is [[inscribed figure|inscribed]] in a curve when all four vertices of the square lie on the curve. The unsolved [[inscribed square problem]] asks whether every [[simple closed curve]] has an inscribed square. It is true for every [[smooth curve]],<ref>{{cite journal |last=Matschke |first=Benjamin |year=2014 |title=A survey on the square peg problem |journal=[[Notices of the American Mathematical Society]] |doi=10.1090/noti1100 |volume=61 |issue=4 |pages=346–352|doi-access=free }}</ref> and for any closed [[convex curve]]. The only other regular polygon that can always be inscribed in every closed convex curve is the [[equilateral triangle]], as there exists a convex curve on which no other regular polygon can be inscribed.<ref>{{cite journal
 | last = Eggleston | first = H. G.
 | doi = 10.1080/00029890.1958.11989144
 | journal = [[The American Mathematical Monthly]]
 | jstor = 2308878
 | mr = 97768
 | pages = 76–80
 | title = Figures inscribed in convex sets
 | volume = 65
 | year = 1958| issue = 2
 }}</ref>

For an [[inscribed square in a triangle]], at least one side of the square lies on a side of the triangle. Every [[acute triangle]] has three inscribed squares, one for each of its three sides. A [[right triangle]] has two inscribed squares, one touching its right angle and the other lying on its hypotenuse. An [[obtuse triangle]] has only one inscribed square, on its longest. A square inscribed in a triangle can cover at most half the triangle's area.<ref name=gardner>{{cite journal
 | last = Gardner | first = Martin | author-link = Martin Gardner
 | date = September 1997
 | doi = 10.1080/10724117.1997.11975023
 | issue = 1
 | journal = [[Math Horizons]]
 | pages = 18–22
 | title = Some surprising theorems about rectangles in triangles
 | volume = 5}}</ref>
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