# Beautiful Proofs In Geometry **URL:** \ **Category:** In My Humble Opinion\ **Created:** [November 23, 2005, 10:26pm UTC](https://boards.straightdope.com/t/beautiful-proofs-in-geometry/332496 "2005-11-23T22:26:44Z")\ **Posts on this page:** 14\ **Page:** 1
**Author:** ![groman](https://avatars.discourse-cdn.com/v4/letter/g/73ab20/32.png) [@groman](https://boards.straightdope.com/u/groman)\ **Post date:** [November 23, 2005, 10:26pm UTC](https://boards.straightdope.com/t/beautiful-proofs-in-geometry/332496/1 "2005-11-23T22:26:44Z")
I have been tasked with an assignment to come up with a 30 minute lesson in Geometry using a software demonstration (Sketchpad, Cabri, WinGEOM, anything) at a high school/undergraduate level on “something interesting”. Now, I have a vague idea of what I want to do, probably an interactive investigation of extending the idea of a nine-point-circle to some non-euclidian geometry (I haven’t checked yet, for all I know it doesn’t extend to ANY non-euclidian geometry), but that’s just… BORING. You guys know of any truly beautiful proofs that I can graphically demonstrate instead? Something that takes 15-25 minutes to describe in detail to a high schooler? Regards, Groman ---
**Author:** ![Quartz](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/quartz/32/267_2.png) [@Quartz](https://boards.straightdope.com/u/Quartz)\ **Post date:** [November 23, 2005, 10:59pm UTC](https://boards.straightdope.com/t/beautiful-proofs-in-geometry/332496/2 "2005-11-23T22:59:06Z")
How about the Dot Theorem? ---
**Author:** ![groman](https://avatars.discourse-cdn.com/v4/letter/g/73ab20/32.png) [@groman](https://boards.straightdope.com/u/groman)\ **Post date:** [November 24, 2005, 12:09am UTC](https://boards.straightdope.com/t/beautiful-proofs-in-geometry/332496/3 "2005-11-24T00:09:48Z")
> [@Quartz](#): > > How about the Dot Theorem? Pardon my ignorance, but what’s the Dot Theorem? ---
**Author:** ![MaxTheVool](https://avatars.discourse-cdn.com/v4/letter/m/9fc29f/32.png) [@MaxTheVool](https://boards.straightdope.com/u/MaxTheVool)\ **Post date:** [November 24, 2005, 12:29am UTC](https://boards.straightdope.com/t/beautiful-proofs-in-geometry/332496/4 "2005-11-24T00:29:00Z")
A geometric explanation of the pythagorean theorem? That might be too basic, depending on what level of math they’re at, but it’s a hugely basic and important fact with a nice visual explanation which I’ve only rarely seen discussed. A discussion of newton’s method might be interesting, as you have little lines bouncing all over the place, intersecting the axis, etc. This would be particularly interesting if you could come up with some examples in which newton’s method didn’t work and show why it was. ---
**Author:** ![SlowMindThinking](https://avatars.discourse-cdn.com/v4/letter/s/dc4da7/32.png) [@SlowMindThinking](https://boards.straightdope.com/u/SlowMindThinking)\ **Post date:** [November 24, 2005, 12:48am UTC](https://boards.straightdope.com/t/beautiful-proofs-in-geometry/332496/5 "2005-11-24T00:48:59Z")
I don’t know if you have the time, but you could grab some of Newton’s proofs from a translation of the Principia. He used geometry for almost everything, including proving that planets (objects undergoing a central force) move in a plane. Alternatively, you could show that a vector dragged around a globe on some trajectory that returns to the starting point does not necessarily return pointing in the same direction. ---
**Author:** ![Qwertyasdfg](https://avatars.discourse-cdn.com/v4/letter/q/ecc23a/32.png) [@Qwertyasdfg](https://boards.straightdope.com/u/Qwertyasdfg)\ **Post date:** [November 24, 2005, 1:08am UTC](https://boards.straightdope.com/t/beautiful-proofs-in-geometry/332496/6 "2005-11-24T01:08:35Z")
Star pentagram? ---
**Author:** ![NinjaChick](https://avatars.discourse-cdn.com/v4/letter/n/71c47a/32.png) [@NinjaChick](https://boards.straightdope.com/u/NinjaChick)\ **Post date:** [November 24, 2005, 2:29am UTC](https://boards.straightdope.com/t/beautiful-proofs-in-geometry/332496/7 "2005-11-24T02:29:00Z")
Take a browse through Euclid’s [Elements](http://aleph0.clarku.edu/~djoyce/java/elements/elements.html). [Prop 47 of book I](http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI47.html) (the Pythagorean theorem) is awesome. He also does a whole lot (in the middle books, I think it starts around book 5) with ratios and proportions, but does it all geometrically. ---
**Author:** ![Snooooopy](https://avatars.discourse-cdn.com/v4/letter/s/ee7513/32.png) [@Snooooopy](https://boards.straightdope.com/u/Snooooopy)\ **Post date:** [November 24, 2005, 3:07am UTC](https://boards.straightdope.com/t/beautiful-proofs-in-geometry/332496/8 "2005-11-24T03:07:49Z")
> [@Qwertyasdfg](#): > > Star pentagram? That’s right … summon the devil and let HIM give the presentation. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [November 24, 2005, 5:00am UTC](https://boards.straightdope.com/t/beautiful-proofs-in-geometry/332496/9 "2005-11-24T05:00:18Z")
Offer a proof of the following: Given a circle C with center O, a point P on C, and a line _l_ which intersects C at P, _l_ is tangent to C iff _l_ is perpendicular to the radius OP. It can be done using nothing but high school-level geometry and algebra, it should take about the right time to present, and it’s likely to be the most complicated proof the students have seen to date. ---
**Author:** ![Fern\_Forest](https://avatars.discourse-cdn.com/v4/letter/f/71c47a/32.png) [@Fern\_Forest](https://boards.straightdope.com/u/Fern_Forest)\ **Post date:** [November 24, 2005, 6:19am UTC](https://boards.straightdope.com/t/beautiful-proofs-in-geometry/332496/10 "2005-11-24T06:19:51Z")
How about showing how you can use origami to trisect a angle? ---
**Author:** ![Quartz](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/quartz/32/267_2.png) [@Quartz](https://boards.straightdope.com/u/Quartz)\ **Post date:** [November 24, 2005, 6:41pm UTC](https://boards.straightdope.com/t/beautiful-proofs-in-geometry/332496/11 "2005-11-24T18:41:04Z")
> [@groman](#): > > Pardon my ignorance, but what’s the Dot Theorem? A good question: it’s the Dot Conjecture, not Theorem. :smack: ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [November 24, 2005, 7:15pm UTC](https://boards.straightdope.com/t/beautiful-proofs-in-geometry/332496/12 "2005-11-24T19:15:11Z")
So what’s the dot conjecture? Google doesn’t help. ---
**Author:** ![Qwertyasdfg](https://avatars.discourse-cdn.com/v4/letter/q/ecc23a/32.png) [@Qwertyasdfg](https://boards.straightdope.com/u/Qwertyasdfg)\ **Post date:** [November 25, 2005, 6:39am UTC](https://boards.straightdope.com/t/beautiful-proofs-in-geometry/332496/13 "2005-11-25T06:39:00Z")
> [@](#): > > That’s right … summon the devil and let HIM give the presentation. Satan, or Euclid? Come to think of it, I’ve never seen them together at the same time… [http://aleph0.clarku.edu/~djoyce/java/elements/bookIV/propIV11.html](http://aleph0.clarku.edu/~djoyce/java/elements/bookIV/propIV11.html) ---
**Author:** ![Quartz](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/quartz/32/267_2.png) [@Quartz](https://boards.straightdope.com/u/Quartz)\ **Post date:** [November 25, 2005, 7:06pm UTC](https://boards.straightdope.com/t/beautiful-proofs-in-geometry/332496/14 "2005-11-25T19:06:31Z")
> [@ultrafilter](#): > > So what’s the dot conjecture? Google doesn’t help. The Dot Conjecture states that it is impossible to draw a dot diagram such that every line has at least three dots in it. See Simon Singh’s _Fermat’s Last Theorem_, Appendix 6.