# Extremely intuitive geometric proofs for teaching

Does anyone know where I could find a book or resource of very simple intuitive proofs of the basic results in Geometry? I tutor geometry to middle schoolers, and find that due to shoddy mathematical education, they're not used to really rigorous step by step thinking. On the other hand, very intuitive proofs, like the ones from Lockhart's Lament they find very appealing.

If you haven't read Lockhart's Lament: http://www.maa.org/devlin/LockhartsLament.pdf

tl;dr

There's a proof for the area of a triangle that's given by inscribing the triangle in a rectangle. It's immediately obvious why the formula is the way it is that way. There's another nice proof of the theorem about inscribed angles in a circle.

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Well whatever you end up doing, please, please tell me you will teach quadrilaterals correctly. By that I mean that if you define kites and trapezoids, do so with the intention of making rectangles trapezoids and rhombi into kites. You would not believe the number of gradeschool resources that completely miss the point of classifying quads by their properties. Instead they use an unholy hybrid of "by properties" and "by tradition". – rschwieb Nov 2 '12 at 17:27
And thanks for the link to the lament. – rschwieb Nov 2 '12 at 17:30
The geometric proof of Pythagoras's Theorem, or is that not usually taught to middle schoolers? – Rankeya Nov 2 '12 at 20:42
Also, can you give an example of a geometric proof you find non-intuitive? Some might just need you to come up with a clever method, like drawing the right perpendicular line or something of that sort. – Rankeya Nov 2 '12 at 20:48
Hmmm. I personally find most geometric proofs to be pretty intuitive. The problem is that my students often don't understand the reasoning behind proofs that require too many intuitive leaps. Maybe what I'm going for is "simple." – Josh Infiesto Nov 3 '12 at 22:08

## 2 Answers

MAA (Lockhart's Lament source) has two books "Proofs without Words" which cover geometry amongst other things. Conway & Guy "The book of numbers" have visual proofs of various algebraic identities too (beautiful, but not geometry). Coxeter's "Introduction to Geometry" in the early chapters has some stunningly efficient diagrams.

An example I like is that the perpedicular bisectors of the sides of a triangle meet in a point (the circumcentre). The bisector of one side is the collection of points equidistant from its two endpoints. Two such bisectors clearly meet (they aren't parallel). The third must meet at the same point, because the point is already equidistant from the two vertices at the ends of its side.

Now embed your original triangle in a triangle twice as big and upside down, so that the vertices of the original triangle are midpoints of the sides of the bigger one. Then the perpendicular bisectors of the sides of the large triangle meet in a point. But they are also the altitudes of the small triangle.

The first time I saw that, it was a piece of magic.

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I know the spelling is English rather than American, but the attempted autocorrection of "circumcentre" to "circumventer" is the most irritating thing I have yet encountered in dealing with this site. – Mark Bennet Nov 2 '12 at 21:22
Extremely cool. The perpendicular bisectors part is actually a good example. It took my students a while and a lot of prodding to see why the third bisector would meet at the same point. – Josh Infiesto Nov 16 '12 at 19:30

Very glad I remembered this: http://www.amazon.com/Geometry-Imagination-David-Hilbert/dp/0828410879. I am not sure if it is appropriate for all middle-schoolers, but perhaps this is a good book to suggest to more inquisitive ones.

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