Using GeoGebra to Find Product of Chords
Using GeoGebra to Find Product of Chords
Using GeoGebra to Find Product of Chords: UPDATE 1
Sam J. Shah posted the following problem about product of chords on his blog:
Consider the unit circle x2+y2=1. Plot n equally spaced points on the circle starting from (1,0). Now draw the n-1 chords from (1,0) to the others. What is the product of the lengths of all of these chords?
… and offered some cases which he and others verified on twitter and suggested that GeoGebra may be perfect for this.
So I followed up on his suggestion. I thought that it would be a good opportunity to learn how to make an applet using GeoGebra. I’m also partially motivated because, by sheer serendipity, I “attended” a workshop session by Linda Fahlberg-Stojanovska hosted by The First North American GeoGebra Conference at Ithaca College and reminded of a video of hers that I watched a while ago.
The pause button was a friend while I watched the video. It was definitely a little fast for a newbie like me. I must say, though, that it is very well made and I would recommend her channel and the GeoGebra Channel to anyone interested in learning GeoGebra.
While working on this applet, I wished that autocomplete on GeoGebra had a tooltip telling me what parameters the commands took. Google searches for commands were not particularly useful. It also took a little digging through the online reference to find what I needed. I’ve been spoiled by PHP’s HTML Help File in chm (Compiled HTML File) format with index, search, syntax, and code samples.
The final product below do verify Sam’s test cases. Click on the image below to go to the GeoGebra applet. I’ve allowed commands and toolbar for now. Drag the slider to increase or decrease the number of chords. Click on play to animate it.
I love the times we live in. We got tons of tutorials, workshops, references, and videos at our disposal. This is the first applet that I’ve ever made in GeoGebra and I’ve only started working on it after Sam tweeted about it. In the old days, I probably would’ve gone to a bookstore to buy a book or a library to see if a software’s reference/tutorial books were available then I would make my way to a computer lab to learn the basics of a language/software package and tried a few basic examples/hello world programs before I attempted the applet.
YAY for technology and for all the amazing people out there sharing their knowledge with the world. Thanks also go to Sam for sharing a problem that started me on this. I feel much more motivated to learn to use GeoGebra in the classroom now.
UPDATE: Be sure to check out the comments on Sam’s original post. This post was more about learning GeoGebra and the myriad of resources available for GeoGebra than it is about the problem.
UPDATE2: Changed the wording regarding solutions.
This is awesome! Your applet is awesome!
BTW, though, I didn’t actually come up with a solution and post it. Some people posted their thoughts in the comments. But I was getting — via “I’ve shown it true for the first four cases and some random middling cases I checked so it must be true for the rest” — the same answer as from the applet you constructed.
It’s quite amazing, no? The answer? I still don’t *know* why it works out – not intuitively, or even algebraically.
I think that’s one way to tackle a problem. Identifying solutions and non-solutions and begin to look for patterns. The harder part of this problem is explaining why the pattern exists or holds and to “get it” intuitively.
It is quite amazing. Just to be sure I also tried increasing the precision (under Options->Rounding then pick something with more decimal places) on the applet to see how well the conjecture holds up (I began to see rounding errors). I’ve also verified the solutions for 1, 2, 3 chords by hand. Maybe if I spent a little more time here I might begin to see a pattern.
Regarding the sentiment, you’re not alone. This SMBC (Saturday Morning Breakfast Cartoon) sums up how I feel.
Nicely done. Did I read on Twitter that you consider yourself a GeoGebra noob? Man, you gotta be kidding. Awesome!
@David Cox
Yup. First applet. Before this I’ve only used it for graphing, and I used wolfram alpha or MathGV instead of GeoGebra.
It just looks impressive because of animation. But if you watch the video above you can see why it wasn’t too bad making it.
I see potential here. GeoGebra does what those other software/service do plus I can create applets that are dynamic. Definitely gonna spend more time learning it. It helps that I’m stumbling across tons of resources, your blog being a big reason I started to look into it.
Glad to help. For some reason, somethingstrange with GGB seem intuitive. But the sequences and lists aren’t for me. I’m going to have to really look into that because I really like this applet. I was going to work on it, but I don’t think I can come close to improving upon what you’ve already done.
@David Cox
I don’t know them well either. I could follow the example in the video. In the places where she used the center of the circle as a segment’s endpoint I used (1,0). That’s probably the only difference. After that the only thing I added is product[].
Thanks for submitting this to the carnival. To those who commented, you may also want to check my GeoGebra tutorial series here:
http://math4allages.wordpress.com/geogebra/