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Homemidpoint

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# midpoint

If $AB$ is a segment, then its *midpoint* is the point $P$ of the segment whose distances from $B$ and $C$ are equal. That is, $AP=PB$.

The midpoint of segment $AB$ can be found with ruler and compass as follows: Draw two circles with radius $AB$ and centers $A,B$ respectively. Let $P,Q$ the intersection points of the circles. Then the intersection $T$ of $PQ$ wih $AB$ is the midpoint of $AB$.

There are several arguments to see why $T$ is indeed the midpoint of $AB$. Because of the circles having the same radius, $AP=AQ=BP=BQ$. It follows that $\triangle PAQ\cong\triangle PBQ$ and $\triangle BAQ\cong\triangle BAQ$ and that they all are isosceles. Then $\angle APT=\angle TPB$ and so $PT$ is the angle bisector of an isosceles triangle and thus also a median. We conclude that $T$ is the midpoint.

An alternative (yet essentially equivalent) argument is that since $AP=AQ=BP=BQ$, then $ABCD$ is a parallelogram (in fact, a rhombus) and therefore the intersection $T$ of its diagonals is the midpoint of each one.

With the notation of directed segments, the midpoint is the point on the line that contains $AB$ such that the ratio $\frac{\overrightarrow{AP}}{\overrightarrow{PB}}=1$.

Generalization. The notion of a midpoint can be generalized. In a geometry with the congruence axioms (such as a neutral geometry), $P$ is a *midpoint* of points $B$ and $C$ if $P,A,B$ are collinear and line segment $AP$ is congruent to line segment $BP$.

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## Comments

## html vs page images

I'm sorry that this work temproarily only works in page image for now

the problem lies in latex2html not being able to correctly parse pstricks nodes (pst-node) which is a dependency on the package I use for the euclidean drawings.

I'll try to get the picture into a eps soon and include it instead of typing directly the code.

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## Re: html vs page images

Maybe this bug should be forward to Ross Moore?

apk

## Re: html vs page images

I've yet to investigate, since nodes it's a bit advanced topic on PSTricks (latex package to do postcript drawings on LaTeX). I've used pstricks before, check "OrthogoncalCurves" entry for an example. I?ll tell you later what I find

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## [almost solved] Re: html vs page images

the reason for the image not working is because the way pstricks work, didn't reserved space on the output, which was causing dvips -E not to ork correctly.

This is where I found the cause and solution

http://lists.gnu.org/archive/html/fhsst-authors/2004-11/msg00009.html

however, the suggested hints to get rid of the frame didn't work (so I left it for now)

and also.. there's now a bug on l2h which makes the picture not to be cropped correctly , I'm still investigating this (this also happened locally on my machine)

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## Re: [almost solved] Re: html vs page images

Just a thought --- when you have time and if it's not too much trouble, could you consider sharing some of your hard earned wisdom by writing a short document on adding illustrations to Planet Math entries? Maybe it could explain some of the basics of pstricks (enough to make simple geometric drawings), how to upload images so they come out right, suggest some graphics programs that help one edit the images, and the like. I, for one would find it very helpful. Up to now, I've shied away from writing entries which need illustrations, but if someone like you explained the basics needed to make the illustrations and it wasn't excessively difficult, I might start adding illustrated entries and even go back to add some illustrations to existing entries.

## New PM document: Graphics and Planetmath

I think you have a great idea, but I'm better starting a collaborative document at docs section so everybody can contribute, dealing with all graphic-related issues, from how to include an eps, how to convert them from other formats, what programs can *correctly* handle EPS on windows, gnuplot, etc to doing graphics directly with pstricks, xypic and quirks that arise

Most of the external-graphic inclusion is a bit straightforward, but I've been thinking that the part dealign with graphics directly in code would be only efficient havint many many examples (from entries).

Just like I described how to do polar plots with pstricks yesterday, but doing similar stuff for lines, polygons, function graphs, graphs, and geometrical drawings. So if you could compile a list of entries needing illustrations (of many kinds) it'll be great so I can start working the detailed examples. So far the midpoint one and the orthogonal curves are considered, but they deal with some of the more advanced topics, and simpler examples are also appreciated.

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## Re: New PM document: Graphics and Planetmath

Here's the link. I couldn't get it to be a site doc due to tech issues, so I'll maintaint it for now as priv collaboration.

http://planetmath.org/?op=getobj&from=collab&id=53

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## Re: New PM document: Graphics and Planetmath

Before you go to far with this, check out the Graphics section of the PM Content and Style guide. It includes a short description of various graphics packages and how to included eps files in PM entries. Due to some problems the guide is not visible unless you click on Edit and then Render.

## Re: New PM document: Graphics and Planetmath

Yes I did, notice however that there is no problem having such document, since the whole graphic creation and handling is a broad subject. Also, suppose somebody is looking for help or information due to some pics problem, is not so clear the information is to be sought on the content and style guide, thus the need for the document on graphics

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## Re: New PM document: Graphics and Planetmath

I was concerned about duplication of effort. The relevant parts of the guide can pretty much be copy-pasted into the new document.

## Re: New PM document: Graphics and Planetmath

I think some duplication is good; people do not read with perfect recall. Perhaps appropriate portions can be merged back into the general guide periodically.

apk

## Re: New PM document: Graphics and Planetmath

I was reading the doc, but it onlly talks briefly, I was meaning a doc describing in details options (for instance, \includegraphics has many options) and having worked examples for xypic and pstricks

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