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Phoebe 16:46, 11 June 2011 (UTC) Welcome! Please feel free to comment on my page and thank you for your support!


Response to Checklist

You've done a great, very through job on this page. My biggest concern is actually that the page is SO long. Have you thought of taking your section on Pappas chain and making it it's own page? I think you have nearly all of the content for another page here, and moving that would cut this page down significantly. Let me know how you feel about this--I do think it would be a good idea.

Thank you for the comments!
I agree with you. It is really long. I guess I just wanted to be thorough and it turned out that I put on too many stuff. It is a great idea to make Pappus Chain into a page. I'm with you on this point but I don't think I have time to write more stuff about this new topic. I think I'll leave it for other people to add more. What do you think?

You'll see my comments below for other edits. I have yet to do a very thorough reading of the Mohr's circle, so more comments on that will come later. Don't hesitate to ask questions about my comments! AnnaP 7/12

I've done a second round of edits. Fixes this time around should be quicker. It's very close to done!

Thank you, Anna!

Messages to the Future

  • Made three suggestions to the future editors.

References and footnotes

  • Original sources of "borrowed" images are marked if you click them. Most of the images in this page is created by me.
  • Direct quotes are cited.
  • References are listed with links at the bottom of the page.

Good writing


  • I think the topic of this page "Arbelos" is interesting. The main image is appealing. The "More Mathematical Explanation" is comprehensive. Arbelos is widely used in many industries, which you can see in the "Why interesting" section.

Quality of prose and page structuring

  • The beginning paragraph defines arbelos and provides its founder and history.
  • Each section is related to the main topic, and you can see them in other websites that talks about arbelos too.
  • Subsections in the mathematical explanations are listed from easy to hard, from fundamental to expanding. I hide almost every proof and just show the statements, in case people don't want to know the proof but just the properties themselves. The heaviest math is at the very last bottom of the page, the Mohr's Circle.
  • This sentence "Same solution for the right twin circle." could be clearer. Instead, you might want to say something like "because of the symmetry of the arbelos, we can go through this same solution for the right circle."
Phoebe 02:32, 13 July 2011 (UTC)Yeah, I just don't know how to say it in a professional way. Fixed it~
  • I would break up this "We have three equations all representing the area of  \triangle ABC and we know that  Area = \frac{Area^2}{Area} = \frac{(Eq.2)^2}{Eq.3} = Eq.1. "
How about making it more like this: We have three equations all representing the area of  \triangle ABC , so we will combine them. Since  Area = \frac{Area^2}{Area} , we can say that  Area= \frac{(Eq.2)^2}{Eq.3} = Eq.1. "
Phoebe 02:32, 13 July 2011 (UTC)I like it. Fixed it.

Integration of images and text

  • Every image is referred in the context. In every image, the denotations are noted and readers know what each symbol means.
  • Readers are clear about which picture they should look at while viewing this page. Sometimes they can get inspired by the picture and know how to prove the problem.
  • There are no large chunks of words. Even if there are, I think most people won't find them annoying.
  • Can you add an image to go along with your Rectangles section?
Phoebe 02:35, 13 July 2011 (UTC) I had a picture there: Figure 1. Do you think I should add another picture to stand out the rectangle?
Yes, it would be helpful to have a picture right next to the text, even if its the same picture as before repeated AnnaP 15:28, 15 July 2011 (UTC)
I added the picture. Good? Phoebe 16:44, 15 July 2011 (UTC)

Connections to other mathematical topics

  • There are several links to other mathematical topics in or outside of Math Images.

Examples, Calculations, Applications, Proofs

  • I think the equations, calculations, and examples are introduced and made clear to readers.
  • Every statement or property has its proof. Some even are proved in different ways in order to show readers a different point of view.
  • I framed some proofs in terms of how someone might begin to write a proof and tell them how to find clues.
  • Some calculations may seem long, but I'm just trying to make every step clear. I'll change them if many think there are too many calculations.


  • I've noticed that you tend to provide a bit more guiding text with your longer calculations rather than your shorter ones. Along those lines, add a bit more text explaining your arc length calculation. You should also specifically direct the reader to look at the image that's there to help (eg "As we can see on the image on the right, the length of the small semi circle is..."

Phoebe 02:45, 13 July 2011 (UTC) Add more explanation to the Arc Length section and Rectangles section. Is there anywhere else I show put more guiding text?
I think you're okay now. AnnaP 15:30, 15 July 2011 (UTC)
  • Do you have a proof of this "According to a property found by Pappus, the altitude to the base AB of \triangle{ABC} is two times the radius of circle C. height = 2R "? If there's a proof somewhere else on the internet, it would be great to provide a link to that--that way you wouldn't have to do it out yourself.
Phoebe 02:45, 13 July 2011 (UTC) I tried to find it before but I don't know the name of this property, which makes it harder to find. I'll keep trying.
Okay. If you can't find it, put a request for someone else to find it in the notes to the future AnnaP 15:29, 15 July 2011 (UTC)
Okay. Still can't find the property... I put a request at the bottom. Phoebe 16:40, 15 July 2011 (UTC)

Area proof

  • Make sure that you label equation 1 as "We want to show" this. It should be crystal clear to the reader that you are working to show this. At the beginning of that section, it sounds like you're assuming that the arbelos area is equal to the area of a circle. That isn't at all transparent, so try to reword that a bit.
Phoebe 03:33, 13 July 2011 (UTC)Yeah, that sentence is confusing so I deleted it.
  • Make sure that you give the equation for the area of a semi circle. So, after you say "The area of a circle is  \pi \frac{d^2}{4} ," add on "therefore the area of a semicircle is \frac{1}{2} \pi \frac{d^2}{4} "
Phoebe 03:11, 13 July 2011 (UTC) Fixed it.
  • During this proof, you start your equation labeling over again. That's very confusing for the reader. There should only be one "equation 1" on the entire page. Make sure that the labeling works throughout the page.
Phoebe 03:11, 13 July 2011 (UTC) Oh, I didn't notice there are two Eq.1's! Sorry.

Mathematical Accuracy and precision of language

  • I try to make everything as clear as possible. Hopefully readers with any level of math will understand it.
  • I try to make everything error free. Corrections and suggestions are appreciated.
  • The definition of every mathematical term, theorem or rule is either explained in the body text or via a mouse-over, or linked to another page.
  • Can you add a definition of semiperimeter? If I don't know what it is off the top of my head (I can figure it out, but it's not 100% obvious), I doubt most people will.
Phoebe 03:34, 13 July 2011 (UTC)I changed it into " s is half of the the perimeter of the triangle" instead of leaving the confusing "semiperimeter" there.


  • Texts are short, not very long, and broken up by images or broken in paragraphs.
  • Mathematical terms are boldfaced. Hide and see is appropriately used.
  • No awkward white chunks.
  • No weird computer codes.
  • Just the very last part "Mohr's Circle" may have chunks of equations and calculations.
  • During your area proof, try to add a few extra line breaks when you are about to start your proof that BD^2=AB*BC
Phoebe 03:36, 13 July 2011 (UTC) Agree! Fixed it. Better?
  • You might want to hide the algebra in the Twin circles section, but it is okay as is.
Phoebe 03:36, 13 July 2011 (UTC) You mean hiding the algebra in the three steps? That's what I did at the beginning but then some one left me comments and told me that she wanted to see the prove directly.
  • You need to add some spaces in here: Therefore u_1^2 + u_2 ^2 + u_3 ^2 = 1 v_1 + v_2 + v_3 = 1 . This equations represents a triangular plane in \mathbf{R^3}. See the right figure. AnnaP 15:35, 15 July 2011 (UTC)
I rearranged this part a little bit. Are they better? Phoebe 16:53, 15 July 2011 (UTC)
  • Also, in the Mohr's circle section, Figure 8 breaks up the text awkwardly in my browser. Can you make sure that figure 8 is inserted after the equations and next to the text that refers to it? AnnaP 15:35, 15 July 2011 (UTC)
I see. Those equations are aligned together so I can't break them apart. Figure 8 was under the equations before, but it kinda got in the way of the next section, which is why I put it before the equations now. Didn't realize that it looks different in other browsers. I changed it back, next to the text. What do you think of it now? Phoebe 16:53, 15 July 2011 (UTC)

Thank You

Thank everyone who has helped me with this page! We are all math lovers!!!

General comments

  • Kate 18:47, 28 June 2011 (UTC): Be careful about your image title being different than your page title. When the page is live, it's the image title that shows up under it in the thumbnail galleries and stuff. Since your initial caption (the words prior to the TOC) explain that the image is a knife in the shape of an arbelos, I'd suggest just changing the image title (the part that now says "A Head Knife") to "Arbelos" to match your page title.
Phoebe 20:16, 28 June 2011 (UTC)Hmm...Never thought of that. Good point. I've changed the title. BTW, what is TOC?
Kate 23:33, 29 June 2011 (UTC): TOC = Table of Contents, sorry!
  • Kate 19:54, 28 June 2011 (UTC): I thought this page was great! :)
Phoebe 20:16, 28 June 2011 (UTC)Thank you!
Phoebe 21:49, 28 June 2011 (UTC) Overall, great comments! You wrote a lot and they are all very helpful to me!! I can't believe you read the whole page so carefully and left those wonderful comments only after a few hours we skyped!
Kate 23:33, 29 June 2011 (UTC): It's no problem. I only hope it wasn't too intimidating for everyone that I left so many comments… I know looking at a discussion page that's all one color can be daunting.
Phoebe 15:36, 30 June 2011 (UTC) As a matter of fact, I think you are just doing your job and helping other people. It is not intimidating at all.

Section-specifc comments

Basic Description

  • Kate 18:47, 28 June 2011 (UTC): I think this section is really great! It's simple and clear. The only thing I might suggest is moving some of the initial caption down here - a lot of readers will miss information that's above the TOC.
Phoebe 20:16, 28 June 2011 (UTC)Fixed it.

A More Mathematical Explanation


  • Kate 18:47, 28 June 2011 (UTC): I don't think the sentence here should be bolded.
Phoebe 20:16, 28 June 2011 (UTC)Fixed it.

Arc Length

  • Kate 18:47, 28 June 2011 (UTC): Since there's just the one proof in this section, I don't think it needs to be hidden - once I've clicked to expand the Arc Length section, I should be able to see the proof too. Also, try and offer a couple sentences more of explanation for how you get from one step to the next in the proof.
Phoebe 20:16, 28 June 2011 (UTC)Fixed it.


Kate 18:47, 28 June 2011 (UTC):

  • You might need to remind people how we know that triangle ACD is right.
Phoebe 20:16, 28 June 2011 (UTC)Fixed it.
  • ( Here, the name of the line segment in parentheses indicates the length of the segment.)
You can make the line segment symbol by typing \overline{AB} (looks like  \overline{AB})
Phoebe 04:14, 8 July 2011 (UTC) Well... There are a lot to change, so I think I'll leave them that way...
  • If you don't know how to prove it, start with what you already know: the area of the circle is the same as the area of the arbelos.
I'd suggest rewording this to "If you don't know how to prove something, start…" just to make it more general - that way it reads as advice that's useful in other situations too. (This is a super minor suggestion, it's fine as is.)
Phoebe 20:16, 28 June 2011 (UTC)Fixed it.
Actually, this looks like it's the same as it was, but like I said, it was just a minor suggestion, not a correction or anything, so that's fine! -Kate 23:51, 29 June 2011 (UTC)
  • If you simplify this "big" equation, you will find that you only need to prove that (BD)2 = (AB)(BC). Here we translate the seemingly sophisticated problem we are going to prove into something nice and simple. Reduction to absurdity is a very important method when dealing with mathematical problems when you don't have a clue.
I think "Here we translate the…" is a little awkward - I think it would sound better as "We have now translated the seemingly sophisticated problem of proving that the area of the circle was the same as the area of the arbelos into something nice and simple." (The first change is to make it better English, the second is to remind the reader of what our original really big problem was.)
Phoebe 20:16, 28 June 2011 (UTC) Agree. Fixed it.
I don't think the sentence that begins "Reduction to absurdity…" should be bolded - I agree that it's important, but it looks a little weird like that. Also, it might read better if the second "when" was a "where".
Phoebe 21:40, 28 June 2011 (UTC) Fixed the second "where". About the bolded sentence, I wanted to point it out because it's more like a general suggestion. I agree that it is weird and this second proof looks messy. I will fix it later but I do want to show its importance.
Kate 23:36, 29 June 2011 (UTC): It definitely is a key sentence, and a general suggestion, but it might stand out well enough without being bolded. I don't know - I'd get some other opinions, but it's up to you.
I really like this bit - I think it's really great that you're framing this proof in terms of how someone might begin to write a proof. I even think it might be good to make this the first proof under this section, so that if someone reads only one proof, this is the one they read.
Phoebe 21:44, 28 June 2011 (UTC) Great idea!!! I totally agree with you. Yeah, I want to make it more illuminative, so I showed the readers how to write a proof, where should you begin instead of just telling you this is the proof.
Kate 23:36, 29 June 2011 (UTC): Yeah, and I think you did a really good job of writing it as a journey instead of just a declaration, it's much more fun for the reader that way.
  • Now back to the property that we need to prove, the following lines demonstrate why (BD)^2 = (AB)(BC):
This sentence is a little awkward - I'd say just "Now, let's prove that (BD)^2 - (AB)(BC):"
Phoebe 21:40, 28 June 2011 (UTC) Fixed it.
  • Your spacing is a little weird in this second proof. I don't really have any suggestions for how to make it better, but I thought I'd point it out.
  • The last proof in this section ends a little abruptly. Remind the reader why the statement A=C1 + C2 proves our initial statement.
Phoebe 21:40, 28 June 2011 (UTC) Fixed it.
Yeah, I think this worked really well! :)</s>


  • Kate 19:54, 28 June 2011 (UTC): You might want to be more consistent about when you use math font and when you use regular writing in this section - we have a math prof here who would insist that A should always mean something different than A. Other than that, I thought this section was good! :)
Phoebe 20:48, 28 June 2011 (UTC)Fixed it.


Kate 19:54, 28 June 2011 (UTC):

  • I think this proof could deal with a little more explanation. Remind us that we can prove the line tangent by showing that it's perpendicular to the radius. And I'd try not to use either of the triple-dots symbols (I forget their real names!), not everyone who's interested in understanding this proof will know/remember what they mean.
Phoebe 21:03, 28 June 2011 (UTC) Yeah, I realized that as well. Fixed it!
Changes look good, proof is more readable! -Kate 23:51, 29 June 2011 (UTC)
  • One more thing to point out is that it is not a coincidence that B, E, D, F are on the circle.
A little awkward. I'd try "One more thing: it isn't a coincidence that B, E, D, and F are on the same circle."
I thought this little note was great, by the way - I hadn't even realized that!
Phoebe 20:48, 28 June 2011 (UTC)Fixed it. Thank you!

Archimedes' Twin Circles

Radius, Diameter and Area

Kate 19:54, 28 June 2011 (UTC):

  • I would re-title this section "Proving the circles congruent" or something else with the word "proof" in it, just to be clear about what the purpose of the section is.
Phoebe 21:21, 28 June 2011 (UTC) Fixed it.
  • I don't like your bullet points in this section, I think you can just have regular text.
Phoebe 21:21, 28 June 2011 (UTC) Fixed it.
  • I don't think the computations under 1,2 and 3 should be hidden - once I click to show the proof, I want to see the proof!
Phoebe 21:21, 28 June 2011 (UTC) Fixed it.
  • I think a little more explanation in the second two chunks of computation would be helpful.
Phoebe 21:21, 28 June 2011 (UTC) Fixed it.
Kate 23:56, 29 June 2011 (UTC): I think there could still be a tad more explanation in the third part - my eyes tend to skip over the computation.

Archimedes’ Circles and the Problem of Apollonius

  • Kate 19:54, 28 June 2011 (UTC): I don't like that when I expand this section, the picture shows up under the next section.
Phoebe 21:12, 28 June 2011 (UTC) I agree. Fixed it.

Bankoff Circle

Kate 19:54, 28 June 2011 (UTC):

  • Since they are all identical, the radius of the Bankoff Circle is \frac{r(1- r)}{2}.
Before when you were talking about the twin circles, you talked about them in terms of diameter, it might be a good idea to do one or the other.
Phoebe 21:11, 28 June 2011 (UTC) Good point! Fixed it.
  • I don't think you need the bullet points in this section.
Phoebe 21:11, 28 June 2011 (UTC) I agree with you. Fixed it.

Pappus Chain



Kate 19:54, 28 June 2011 (UTC):

  • for the height proof, be clearer at the start that we're inverting over the nth circle in the chain.
Phoebe 18:49, 30 June 2011 (UTC) Fixed it. May still need more explanation but I don't know how to rephrase.
  • Again, I'd try and avoid using the triple-dots symbols.
Phoebe 18:49, 30 June 2011 (UTC) Fixed it.
  • Your link to Steiner's Porism isn't good, btw. I'm not sure why, it looks like it's the right title, but it's showing up red.
Phoebe 21:28, 28 June 2011 (UTC) Don't know why, but I fixed it.

Pappus Chain and Steiner Chain

Why It's Interesting

  • Kate 19:54, 28 June 2011 (UTC): I thought these applications were all really interesting!
Phoebe 20:16, 28 June 2011 (UTC) Thank you!


Leather cutting

Art & Design

  • Kate 19:54, 28 June 2011 (UTC): Be careful, the picture for this section looks like it's in the next section.
Phoebe 20:16, 28 June 2011 (UTC)Fixed it and I added another picture.
Kate 13:19, 30 June 2011 (UTC): I like that sculpture - where is it located?
Phoebe 15:42, 30 June 2011 (UTC) I only know it's in Netherlands.
Kate 18:34, 6 July 2011 (UTC): That's good enough, gives it some context. I went in and changed "Netherlands" to "the Netherlands" - for some reason, and I don't really know why, the country is always called "the Netherlands" in English - I figured that was a small enough change it'd be easier for me to do it than to tell you to fix it.
Phoebe 04:12, 8 July 2011 (UTC) Good to know~ LOL. Thanks!


  • Kate 19:54, 28 June 2011 (UTC): I had a little bit of a harder time following this than I did your other proofs (despite having taken linear algebra), and I think it's because I know nothing of engineering or mechanics. You might want to go through it again with the goal of making sure it's clear to those who know nothing about this stuff.
Phoebe 18:49, 30 June 2011 (UTC) Will work on it. It took me a whole afternoon to figure it out too! It's really hard to understand. It's gonna be a big revision, so I'm gonna take my time. : )
===Archimedean Circles===
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