Editing Rectangle (section)

==Theorems==
The [[isoperimetric theorem]] for rectangles states that among all rectangles of a given [[perimeter]], the square has the largest [[area]].

The midpoints of the sides of any [[quadrilateral]] with [[perpendicular]] [[diagonals]] form a rectangle.

A [[parallelogram]] with equal [[diagonals]] is a rectangle.

The [[Japanese theorem for cyclic quadrilaterals]]<ref>[http://math.kennesaw.edu/~mdevilli/cyclic-incentre-rectangle.html Cyclic Quadrilateral Incentre-Rectangle] {{Webarchive|url=https://web.archive.org/web/20110928154652/http://math.kennesaw.edu/~mdevilli/cyclic-incentre-rectangle.html |date=2011-09-28 }} with interactive animation illustrating a rectangle that becomes a 'crossed rectangle', making a good case for regarding a 'crossed rectangle' as a type of rectangle.</ref> states that the incentres of the four triangles determined by the vertices of a cyclic quadrilateral taken three at a time form a rectangle.

The [[British flag theorem]] states that with vertices denoted ''A'', ''B'', ''C'', and ''D'', for any point ''P'' on the same plane of a rectangle:<ref>{{cite journal |author1=Hall, Leon M. |author2=Robert P. Roe |name-list-style=amp |title=An Unexpected Maximum in a Family of Rectangles |journal=Mathematics Magazine |volume=71 |issue=4 |year=1998 |pages=285–291 |doi=10.1080/0025570X.1998.11996653 |url=http://web.mst.edu/~lmhall/Personal/HallRoe/Hall_Roe.pdf |jstor=2690700 |access-date=2011-11-13 |archive-date=2010-07-23 |archive-url=https://web.archive.org/web/20100723134734/http://web.mst.edu/~lmhall/Personal/HallRoe/Hall_Roe.pdf |url-status=dead }}</ref>
:<math>\displaystyle (AP)^2 + (CP)^2 = (BP)^2 + (DP)^2.</math>

For every convex body ''C'' in the plane, we can [[Inscribed figure|inscribe]] a rectangle ''r'' in ''C'' such that a [[homothetic transformation|homothetic]] copy ''R'' of ''r'' is circumscribed about ''C'' and the positive homothety ratio is at most 2 and <math>0.5 \text{ × Area}(R) \leq \text{Area}(C) \leq 2 \text{ × Area}(r)</math>.<ref>{{Cite journal | doi = 10.1007/BF01263495| title = Approximation of convex bodies by rectangles| journal = Geometriae Dedicata| volume = 47| pages = 111–117| year = 1993| last1 = Lassak | first1 = M. | s2cid = 119508642}}</ref>

There exists a unique rectangle with sides <math>a</math> and <math>b</math>, where <math>a</math> is less than <math>b</math>, with two ways of being folded along a line through its center such that the area of overlap is minimized and each area yields a different shape{{Snd}}a triangle and a pentagon. The unique ratio of side lengths is <math>\displaystyle \frac {a} {b}=0.815023701...</math>.<ref>{{cite OEIS|A366185|	Decimal expansion of the real root of the quintic equation <math>\ x^5 + 3x^4 + 4x^3 + x -1 = 0</math> }}</ref>