I am teaching euclidean geometry this fall and realized i dont know it that well. there are some famous modern versions of the axioms which do not completely satisfy me, such as hilberts, gasp. i said it. i especially like the new book by hartshorne, geometry euclid and beyond, because he makes it so clear that i can choose to disagree with him and still he covers my point of view too. e.g., i think it less natural to choose pasch's theorem as a postulate (a line meeting a triangle away from a vertex meets it again.) than simply to say a line separates the plane into two disjoint sides. i also think it unintuitive to postulate that SAS implies congruence as so many books do now in high school, such as the previously wonderful book of jacobs. it seems more in keeping with euclid to postulate instead a set of rigid motions, and use them as he did to prove SAS as a theorem. I mean I think the most natural axioms are euclids stated ones plus his unstated ones that he used in his proofs. this to me is genuinely euclidean geometry fortunately hartshorne proves that one can substitute rigid motions for the SAS postulate he himself uses, and it is easy to show that Pasch can be replaced by plane separation. but i suggest high school books could benefit from such changes too. on the other hand high school books actually do not always even mention the fact that a line separates the plane into two sides, as if it is better left unsaid. without it however, one might be working in three space! i am taking an intuitive approach to geometry at first and hoping to gradually introduce more precise language and proof at the middle or end. do you recall such theorems as the incidence results for medians, altitudes, perpendicular bisectors? angle bisectors? and their link with inscribing or circumscribing circles about triangles? does everyone (except me) know that the angle cut by a pair of rays with vertex on a circle, does not depend on the location of the vertex? just of the two other intersection points with the circle? i.e. any pair of rays with vertex on a circle subtends an angle of exactly half the arc cut from the circle by the two rays. more people probably know that any angle with vertex on a circle, and with rays cutting the circle at the ends of a diameter is a right angle. it is also easy without calculus to compute both surface area and volume of a sphere, (finessing limits), using cavalieris principle. it seems actually easier and clearer to explain them this way than to use calculus. im beginning to find this stuff interesting. then too the contrast betwen euclidean and spherical and hyperbolic geometry is interesting. what do you guys think?