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# cyclic quadrilateral

Cyclic quadrilateral.

A quadrilateral is cyclic when its four vertices lie on a circle.

A necessary and sufficient condition for a quadrilateral to be cyclic, is that the sum of a pair of opposite angles be equal to $180^{\circ}$.

One of the main results about these quadrilaterals is Ptolemy’s theorem.

Also, from all the quadrilaterals with given sides $p,q,r,s$, the one that is cyclic has the greatest area. If the four sides of a cyclic quadrilateral are known, the area can be found using Brahmagupta’s formula

Related:

OrthicTriangle, PtolemysTheorem, ProofOfPtolemysTheorem, Circumcircle, Quadrilateral

Synonym:

cyclic

Type of Math Object:

Definition

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

51-00*no label found*81R50

*no label found*81P05

*no label found*81Q05

*no label found*81-00

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## Comments

## geometric diagrams

How do you make geometric diagrams (like the one in the cyclic quadrilateral article)? I'm new to TeX and would like to know how to do that.

Wkbj79

## Re: geometric diagrams

There are basically two approaches for graphics in TeX.

1) you draw them with some external utility (for example xfig) and then you include them in the TeX file as encapsulated postscipt (eps) using

the command \includegraphics

2) you can draw them within TeX possibly by using additioal packages like pstricks or xypic.

[Actually there is also

1.5) use some external utility (like dia, xeukleides, DrGenius,...) to draw the pictures and then have that utility output some LaTeX code (usually using pstricks) which you then insert in to your document.]

I personally favor option 2 (or when I feel lazy 1.5) but that is largely a matter of personal taste. Actually pstricks is very powerful

you can almost do whatever postscript can do. You can check documentation and a lot of examples at

http://www.tug.org/applications/PSTricks/

By the way you can always check the LaTeX source of any of the entries to see how the author did a particular thing that you want to imitate.

When you chose display method one of the options is "TeX" source. If you try something and it doesn't work then check the preamble of the document, it may have some user defined commands or some nonstandard package. IMHO that is the best way to learn LaTeX.

## Re: geometric diagrams

Which program is the best for PSTricks in windows

Thanks

## Re: geometric diagrams

I am not sure I understand this question.

PStricks is a latex package and I think it comes with most TeX distributions. To see if it is in your system write a document with the following somewhere in its preamble:

\usepackage{pstricks}

If it compiles then it is installed in your system. Otherwise you will have to install it yourself. Then you can use your ordinary editor that you use to edit TeX files.

But maybe you are asking what program best outputs PStricks code under windoze. I don't know (I don't use windoze). I know that many unix/linux applications have been ported to windoze by cygwin.

## Re: geometric diagrams

I personally like to use matlab and Adobe Illustrator to make .eps postscript files and just include them in.

## Re: geometric diagrams (I did it this way)

here's how I did that pic

there's a program called mayura draw (for windows)

that lets you export to eps

(I found exporting to ps/eps on windows is kinda hard,

and most programs either produce innecesarily huge files (~1mb) or drawing loses quality

I've found that gimp for iwndows does well, but gimp is unsuitable for heavy geometric drawing. So I found mayura. Bad thing is shareware, but hey! they got an old version (for win3.1) calle pagedraw which works perfectly (even on win32) and it's free, I used that for the drawing.

Later I moved to unix enviroments, and I started with xfig, but I dislike some features of it, so my current tool is a program callex xeukleides, an example is on "desargues theorem" entry

http://planetmath.org/encyclopedia/DesarguesTheorem.html

xeukleides is like a tiny programming language (with natural geometric instructions like line, circle, itnersection, height, triangle, etc)

and you can "compile" your code into an eps

on that page there's a link to a sampel source code

I think that can be ported to windows (with cygwin) since it's small, but I never got enough motivation/skills

f

G -----> H G

p \ /_ ----- ~ f(G)

\ / f ker f

G/ker f