Equidiagonal quadrilateral

Examples of equidiagonal quadrilaterals include the isosceles trapezoids, rectangles and squares.

File:Reuleaux kite.svg]]

Among all quadrilaterals, the shape that has the greatest ratio of its perimeter to its diameter is an equidiagonal kite with angles π/3, 5π/12, 5π/6, and 5π/12.{{citation

|first=D.G.

|last=Ball

|title=A generalisation of π

|journal=Mathematical Gazette

|volume=57

|issue=402

|year=1973

|pages=298–303

|doi=10.2307/3616052|jstor=3616052

}}, {{citation

|first1=David

|last1= Griffiths

|first2=David

|last2=Culpin

|title=Pi-optimal polygons

|journal=Mathematical Gazette

|volume=59

|issue=409

|year=1975

|pages=165–175

|doi=10.2307/3617699|jstor= 3617699

}}.

A convex quadrilateral is equidiagonal if and only if its Varignon parallelogram, the parallelogram formed by the midpoints of its sides, is a rhombus. An equivalent condition is that the bimedians of the quadrilateral (the diagonals of the Varignon parallelogram) are perpendicular.

A convex quadrilateral with diagonal lengths $p$ and $q$ and bimedian lengths $m$ and $n$ is equidiagonal if and only if{{rp|Prop.1}}

:$pq=m^2+n^2.$

The area K of an equidiagonal quadrilateral can easily be calculated if the length of the bimedians m and n are known. A quadrilateral is equidiagonal if and only if{{citation

| last = Josefsson

| first = Martin

| journal = Forum Geometricorum

| pages = 17–21

| title = Five Proofs of an Area Characterization of Rectangles

| url = http://forumgeom.fau.edu/FG2013volume13/FG201304.pdf

| volume = 13

| year = 2013

| access-date = 2013-02-09

| archive-date = 2016-03-04

| archive-url = https://web.archive.org/web/20160304001152/http://forumgeom.fau.edu/FG2013volume13/FG201304.pdf

| url-status = dead

}}.{{rp|p.19;}} {{citation

| last = Josefsson

| first = Martin

| journal = Forum Geometricorum

| pages = 129–144

| title = Properties of equidiagonal quadrilaterals

| url = http://forumgeom.fau.edu/FG2014volume14/FG201412index.html

| volume = 14

| year = 2014

| access-date = 2014-08-28

| archive-date = 2024-06-05

| archive-url = https://web.archive.org/web/20240605032351/https://forumgeom.fau.edu/FG2014volume14/FG201412index.html

| url-status = dead

}}.{{rp|Cor.4}}

:$\backslash displaystyle\; K=mn.$

This is a direct consequence of the fact that the area of a convex quadrilateral is twice the area of its Varignon parallelogram and that the diagonals in this parallelogram are the bimedians of the quadrilateral. Using the formulas for the lengths of the bimedians, the area can also be expressed in terms of the sides a, b, c, d of the equidiagonal quadrilateral and the distance x between the midpoints of the diagonals as{{rp|p.19}}

:$K=\backslash tfrac\{1\}\{4\}\backslash sqrt\{(2(a^2+c^2)-4x^2)(2(b^2+d^2)-4x^2)\}.$

Other area formulas may be obtained from setting p = q in the formulas for the area of a convex quadrilateral.

A parallelogram is equidiagonal if and only if it is a rectangle,{{citation

| last = Gerdes

| first = Paulus|author-link=Paulus Gerdes

| issue = 2

| journal = Educational Studies in Mathematics

| jstor = 3482571

| pages = 137–162

| title = On culture, geometrical thinking and mathematics education

| volume = 19

| year = 1988

| doi = 10.1007/bf00751229}}. and a trapezoid is equidiagonal if and only if it is an isosceles trapezoid. The cyclic equidiagonal quadrilaterals are exactly the isosceles trapezoids.

There is a duality between equidiagonal quadrilaterals and orthodiagonal quadrilaterals: a quadrilateral is equidiagonal if and only if its Varignon parallelogram is orthodiagonal (a rhombus), and the quadrilateral is orthodiagonal if and only if its Varignon parallelogram is equidiagonal (a rectangle).{{citation

| last = de Villiers | first = Michael

| isbn = 9780557102952

| page = 58

| publisher = Dynamic Mathematics Learning

| title = Some Adventures in Euclidean Geometry

| url = https://books.google.com/books?id=R7uCEqwsN40C&pg=PA58

| year = 2009}}. Equivalently, a quadrilateral has equal diagonals if and only if it has perpendicular bimedians, and it has perpendicular diagonals if and only if it has equal bimedians.{{citation

| last = Josefsson

| first = Martin

| journal = Forum Geometricorum

| pages = 13–25

| title = Characterizations of Orthodiagonal Quadrilaterals

| url = http://forumgeom.fau.edu/FG2012volume12/FG201202.pdf

| volume = 12

| year = 2012

| access-date = 2012-04-23

| archive-date = 2020-12-05

| archive-url = https://web.archive.org/web/20201205213638/http://forumgeom.fau.edu/FG2012volume12/FG201202.pdf

| url-status = dead

}}. See in particular Theorem 7 on p. 19. {{harvtxt|Silvester|2006}} gives further connections between equidiagonal and orthodiagonal quadrilaterals, via a generalization of van Aubel's theorem.{{citation

| last = Silvester | first = John R.

| issue = 517

| journal = The Mathematical Gazette

| jstor = 3621406

| pages = 2–12

| title = Extensions of a theorem of Van Aubel

| volume = 90

| year = 2006| doi = 10.1017/S0025557200178969

}}.

Quadrilaterals that are both orthodiagonal and equidiagonal are called midsquare quadrilaterals because they are the only ones for which the Varignon parallelogram (with vertices at the midpoints of the quadrilateral's sides) is a square.{{rp|p. 137}}

{{reflist}}

{{Polygons}}

Category:Types of quadrilaterals

• [https://dynamicmathematicslearning.com/equidiagonal-quad-van-aubel.html A Van Aubel like property of an Equidiagonal Quadrilateral] at [http://dynamicmathematicslearning.com/JavaGSPLinks.htm Dynamic Geometry Sketches], interactive geometry sketches.