Does this forum include euclidean geometry?

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  • #26
mathwonk
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wow, just opened archimedes and found precisely the statements i have been making to people lately about circles and spheres:

namely he says exactly that (for the pruposes of computing area) a circle is a triangle whose vertex is at the center and whose height is the radius. and (in regard to volume) a sphere is a cone whose vertex is at the center and height is the radius.

it makes me fel very good to have arrived at exactly the perspective of archimedes in these matters, although 2,000 years later, and with a great deal of instruction.

too bad i did not read him as a child, i would have known this 50 years ago.

oh, and archimedes knew the location of the centroid both of a triangle and of a cone, the latter something i just learned a few weeks ago and taught to my class.

and euler computes the values of zeta at even integers, i.e. the sum of the reciprocal powers 1/n^2k, for k up to 13, in his precalculus book, vol 1, page 139. actually he just states them, implying the computations are straighforward.

i have still not arrived at euler's perspective.
 
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  • #27
mathwonk
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i have a geometry question.

i am teaching geometry now and trying to prove as much as possible without using the parallel postulate. we have the congruence theorems, SAS, SSS, ASA, AAS, and the weak exterior angle theorem, open mouth theorem, hypotenuse - side congruence for right tringles, and the fact that the angle sum in a triangle is never more than 180degrees, and we are ready for the concurrence theorems for angle bisectors and medians.

(of course one knows the other two basic congruence theorems, for altitudes and perpendicular bisectors, do not hold without the 5th postulate.)

but i cannot find a book with a proof of these things, or at least not for the medians. the books i have, like greenberg, prove first that there are only two geometries satisfying ll these axioms, short of the parallel postulate, and then give two different proofs of the concurrence theorem ffor medians, one for euclidean geometry and one for hyperbolic geometry, using the poincare model.

now clearly that is not the right way to proceed. one should give a single unified proof just using the axioms that hold for both. any suggestions for a reference that does it right? i have a kind of a vague idea myself but do not know how to implement it.

(i wonder if ceva's theorem works in this generality?)
 
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  • #28
mathwonk
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yes that seems to do it. so greenberg does apparentkly have the proof, but the reference i got from millman parker is to the wrong oart of greenberg.

i.e. the proof of ceva in Greenberg, exercise H6 p. 234, is apparently implied there to work in general using the theorem of menelaus, to prove that the medians are either concurrent or parallel, but it is easy to show they are not parallel.
 
  • #29
mathwonk
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oops, the proofs of cevas thm i am finding use unique parallel lines, or similasrv triangles or areas, which are not elementary concepts (or not true) in neutral geometry.
 
  • #30
mathwonk
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ah yes, the hyperbolic area function does NOT satisy bh/2 for area of a triangle, so the various area proportionality proofs of ceva do not hold. in fact the numbers bh/2 are not the same for different choices of base and height.

presumably ceva is true however???????????
 
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