Non-Euclidean III.36

1. INTRODUCTION. Among the theorems of plane geometry, a privileged position is held by those that are true in neutral geometry, that is, without either assuming or denying the parallel postulate. Such, for example, are the first twenty-eight propositions of Euclid's Elements, including the triangle congruences SAS (I.4), 1 SSS (I.8), ASA (I.26), and the base angles of an isosceles triangle are equal (I.5). Some concur-rence theorems, such as the three angle bisectors of a triangle meeting in a point, are easily proved in neutral geometry (cf. (IV.4)). Others, such as the one pertaining to the three altitudes of a triangle (if two of them meet, then the third also meets in the same point) are true in neutral geometry, but more difficult to prove [4, pp. 369, 387, 399, 430]. On the other hand, the Pythagorean theorem (I.47), proved either by using Euclid's theory of area or by using similar triangles, is irrevocably tied to the Euclidean parallel postulate. There is an analogue in hyperbolic geometry (discovered by Bolyai and Lobachevskii) that says sin c = sin a sin b, where a, b, c are the angles of parallelism associated with the sides a and b and the hypotenuse c of a right triangle [4, p. 406]. There is also a non-Euclidean version using area [10], but its statement is not entirely satisfactory, because the figure on the hypotenuse is made to depend on the figures on the two sides.