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.
 A: 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.
A: 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.
