A Math Forum Webmaster Exchange

Subject: Please help: Bramputa's Theory

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Introduction

Jim, Grace, and I have never met--yet from Grace's inquiring e-mail to webmaster, and from Jim's math teacher training and WWW page-building, our communication ultimately helped develop an Internet education resource.

Reflections from Jim, Grace, and myself separate each of the original eleven e-mails of our three-way exchange, below. Standing independent of the particular content of our conversation, our individual conclusions reveal how we conceive the roles of the webmaster, of the Math Forum, and of the WWW in facilitating education and community.

From Grace to Math Forum webmaster

Date: Thu, 17 Apr 1997 22:25:54 -0500 (EDT)

Dear Sir or Madam:

I would like to request information about Bramputa's Theory. I have done many searches on this topic but no one seems no know of it. I would like to understand how this equation is set up and why it works. If possible, by any means, send it to me at my e-mail address.

Bramputa:
When a quadrilateral is inscribed in a circle, finding the area can be proved by Bramputa's theorem:
Area = the square root of (s-a)(s-b)(s-c)(s-d), where s=the semi-perimeter

There is also a similar problem, in which three sides of any triangle are given and the area is found with yet another equation. Area = the square root of s(s-a)(s-b)(s-c), also where s = semi-perimeter.

However, I have proved this theory, otherwise known as Hero's formula. I think it has similar reasoning, but still have not understood its structure.

I would really appreciate it if someone could send me either the procedure in conjuring this great equation, or more info about it. Thank you for your time.

Yours,
Grace

Richard reflects:
Although we webmasters re-route many math problems to Ask Dr. Math, I decided Grace's inquiry wasn't for this K-12 online service because
  • she clearly stated that she wanted information,
  • she seemed to want to prove the theorem without interactive assistance, and
  • as her inquiry was college-level, Dr. Math might not give it very high priority.
So first I tried my own searches.
Grace reflects:
Frankly, I am not very familiar with the Internet, nor am I picky about where I get my information; but I had figured the best way to approach this problem was to get special one-on-one e-mail assistance. That's why I e-mailed the webmasters at the Math Forum.

From Richard to Grace

Date: Fri, 18 Apr 1997 10:39:23 +0100

Dear Grace,

Thanks for writing us.

I see your problem: I tried two searches for "Bramputa," one at the Math Forum's searcher and another at Yahoo--and neither yielded anything. However, a Math Forum search for the exact phrase "Hero's Formula" yielded some leads in one particular <geometry-pre-college> discussion thread, including a reference to Journey Through Genius, a book of the history and derivation of Hero(n)'s Formula.

However, given this scanty state of WWW affairs for Bramputa's theorem, in particular, I encourage you to post your question to some mailing lists, such as geometry-research (lower volume, but college-level audience) or math-teach (greater volume, but mainly k12 teachers). To subscribe, follow the instructions at "About geometry-research" or "About math-teach," respectively.

Since you've already given a well-elaborated description of your question, I feel it's in fine form to simply post as is.

Let us know how this helps you.

Yours,
Richard

Richard reflects:
At first, I felt the disappointment that Grace must have felt when she looked for WWW resources on this theorem and found nothing helpful. Then I looked at her message again: Grace included not just the statement of the theorem, but a corollary (Hero's Formula), as well. With this corollary as an "exact phrase," I searched again, and managed a few resources that I thought might at least enrich Grace's understanding of this theorem's special case.
Grace reflects:
Richard recommended two very different courses of action in his initial reply--participating in mailing list discussions and searching the WWW for Heron's Formula. His recommending such qualitatively different courses of action did not overwhelm me; I found these choices helpful. The more options in finding the solution to Brahmagupta's, the more I could piece together the relationships between both Heron's Formula and Brahmagupta's Formula. Similarities and answers began to appear in front of my eyes after careful thought and analysis between the two formulas.

From Richard to Grace

Date: Sat, 19 Apr 1997 12:22:30 +0100

Hi again, Grace,

While researching a different question, I happened across another WWW resource you might find interesting in your pursuit of Bramputa's Theorem. Jim Wilson, of the University of Georgia, has a page proving and then exploring the consequences of Hero(n)'s formula.

If you choose not to raise your question in one of the mailing lists I recommended, consider e-mailing him directly, as he seems concerned about the decline of teaching this area of geometry.

Richard reflects:
Grace's cognitive link to Hero's formula offered me another toe-hold into the problem--which I pursued as a hypertextual link during another webmaster investigation, and which led us to Jim.

From Grace to Richard

Date: Sat, 19 Apr 1997 23:16:04 -0500 (EDT)

Thank you so much! I really appreciate it. Take care.

Yours,
Grace

Grace reflects:
If I had used all the resources Richard suggested, my research would have been more complete and time efficient. Nonetheless, contacting the mathematician was my number one priority.

From Jim to Grace

Date: Sun, 20 Apr 1997 12:00:00 -0400 (EDT)

Grace --

The formulas you mention

are both well known and appear in many places in our textbooks or history of mathematics books.

One development of Heron's formula is in a paper on my Web Site.

It is an algebraic development, whereas Heron (or Hero, depending on the translator) used straightforward geometric arguments.

What you are calling Bramputa's Theory--maybe you mean theorem--is often found in History of Mathematics texts but not always with that title. I often assign the problem in my problem solving class: To prove that for a simple quadrilateral inscribed in a circle the area is the square root of (s-a)(s-b)(s-c)(s-d) where a, b, c, and d are the lengths of the sides of the quadrilateral and s is the semi-perimeter.

There are several approaches, but if my students ask for help, I suggest that they draw the diagonals of the quadrilateral, and--using the fact that the vertical angles are equal--apply the law of sines to the triangles to find some relationships and substitutions. The result is a bit tedious but not difficult.

Note that if one of the sides of the quadrilateral is allowed to vary, as it approaches zero the formula reduces to Hero's formula. Thus, if you have proved Bramputa's theorem, then Hero's formula follows as a corollary.

I am not aware of an approach that proves Bramputa's theorem as an extension of Hero's formula.

Good luck with your exploration. Write again, if I can be of more help.

Jim Wilson

Jim reflects:
I am a faculty member in the Department of Mathematics Education and the courses I teach and the web pages I maintain are for prospective and inservice teachers. It was probably the mathematics teacher in me that drove my responses. I responded much as I would to some of my own classroom students.

From Grace to Jim

Date: Mon, 21 Apr 1997 18:37:18 -0400 (EDT)

I do apologize, I was mistaken, it is actually Brahmagupta's Formula! Thank you so much for your help! May I Email you again if I have further trouble?

Grace

Richard reflects:
After Grace cc'ed me on the conversation, I was pretty excited for Grace that a professional mathematics educator was assisting her so thoughtfully. The Math Forum has thought about hosting this kind of "ask a mathematician," "ask a master teacher," or "ask a mentor" dialogue to complement the Ask Dr. Math service's K-12 student emphasis.

From Jim to Grace

Date: Tue, 22 Apr 1997, 11:36:00 -0400 (EDT)

Grace --

By all means, e-mail me any time.

Jim

Jim reflects:
As with my own students, my responses never bring closure to a topic, but rather they are intended to move the student along. In my mental image, Grace was not asking for the completed proof, but only some help in going in a useful direction.

From Grace to Jim

Date: Wed, 23 Apr 1997 18:31:36 -0500 (EDT)

Jim,

Thank you for your previous help, but I still cannot prove the theorem. I have been working at it for a while and been researching as much as I can about Brampagupta's Theorem, nonetheless no success. You mentioned earlier about the class you were teaching and I was wondering if you would have any more clues as to solving this problem. It would really be appreciated if you could do so. Again thanks for everything and I hope to hear from you soon.

Yours,
Grace

Grace reflects:
I didn't mind corresponding with Jim at all; in fact, I found him a great resource. He was a one-on-one assistant who could address my very specific questions about Heron's and Brahmagupta's Formulas immediately, and saved me a lot of hit-or-miss researching.

From Richard to Grace

Date: Thu, 24 Apr 1997 14:33:00 -0 (EDT)

Dear Grace,

I was dismayed to read that, even after enlisting Jim's very able counsel, you had not succeeded in proving the formula--or even found resources to assist you further. So I conducted one more round of searches, feeding into Yahoo's search engine the exact phrase "Hero's Formula," Heron's Formula," and then--once I learned, from these searches, the correct spelling--"Brahmagupta's Formula." Here are the most helpful links that I discovered.

So, if you're still struggling with this proof, I suggest 1) searching for Brahmagupta's Formula either on the WWW (Yahoo turned up nine results, although most were to Eric's glossary, and a couple others were totally garbled) or in print resources; and then 2) writing such as Ted and Eric for further direction, much as you did with Jim.

As always, please do let me know how this helps you.

Richard reflects:
Search engines can be soooo picky--but with the correct spelling of the theorem, I could offer one last set of resources.

From Jim to Grace

Date: Fri, 25 Apr 1997 16:05:39 -0500 (EDT)

Grace--

see http://jwilson.coe.uga.edu/emt725/brahmagupta/brahmagupta.html

Jim Wilson

When I heard from Grace again at this point, I knew that our e-mail correspondence was not enough: e-mail could not convey the graphics and equations from my notes. Therefore, using the materials I had from my problem solving course, I put together the html document referred to above. So with this last message of mine to Grace, I was inviting her to use this page I had just put together. I didn't mention that this URL was a new production--and hoped she hadn't thought that I had withheld it from the previous message!

From Grace to Jim

Date: Fri, 25 Apr 1997 18:39:18 -0400 (EDT)

Thanks ever so much for your help, I really appreciate it!

Yours,
Grace


Conclusions

Richard concludes:
In their initial e-mails, both Jim and Grace provided the kind of elaboration that really assists webmasters to hone in on what we need to do. With respect to Grace, who wrote in with the question, her identifying the corollary of the theorem she sought to prove was the difference between me shrugging my shoulders and me really answering her question by pursuing it from another angle.

I feel that this three-way conversation exemplifies the success webmasters can facilitate. Ultimately, I think the richness of this correspondence reflects the ongoing, sometimes messy, process of learning; facilitates the scaffolding technique of responsive pedagogy; plays beautifully into the hypertextual, well-connected nature of the WWW medium; and illustrates how educators, such as Jim, can draw from the WWW as a learning resource and contribute to it in response to real learning needs.

Grace concludes:
I learned the different options in finding an answer over the Internet in such places as the Math Forum and the WWW. Although it's not always clear how to find the solution in one pass, the whole process was quite useful and in some ways fun.

Jim concludes:
I use the Web to present problems, provide background, and challenge students, who also motivate the Web material I develop myself. After corresponding with Grace, I realized a need for a reference on the details of the theorem that she set out to prove, since I did not find such a Web resource immediately. I may soon put in some notes on a trigonometric development . . .

. . . and one of my undergraduate students came yesterday to show me she had completed the proof. If she writes it up for her class web page, I may add a link to her proof.

Ultimately, through both the process and the product of posting this exchange, we have begun to identify what worked, as well as what didn't; to conceive a better model of webmaster exchanges; and to help the Forum improve its webmaster service.

We also believe that sharing this conversation conveys a few significant points about the Math Forum's services:

  • search engines can help users locate WWW resources exactly if the user has perfect orthography, flawless typing, and a command of engine features such as "exact phrase" searches . . . but sometimes a little human intervention and perseverance helps.
  • in addition to pointing to WWW resources, webmasters can stimulate dialogue with third parties--usually entire mailing lists, such as <math-teach> and <geometry-research>, to which I initially referred Grace, but ultimately, in this case, to Jim, a mathematics educator
  • Forum webmasters can participate in meaningful, extended exchanges with users at-large, much as our "Dr. Math"s can--and the better you elaborate on your understanding of the concern, the better we can help you

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Math Forum * * * * 27 May 1997