By Melvin Hausner
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10 AFFINE GEOMETRY The diligent reader, having read the previous sections, should be warned that the methods therein are capable of handling only certain ideas in Euclidean geometry. These include: 1. ” 2. Ratios of distances along a line. 9. 3. Parallel lines. ” In Fig. 63, it is seen that parallelism may be “defined” in terms of points of division. Thus AB || CD if there is a point P on AD and BC such that AP/PD = BP/PC. ) This may be proved with the help of similar triangles. Technically, these are notions belonging to affine geometry—strictly speaking, real affine geometry, since the real numbers were so basic to the formulation of this geometry.
47), we first mix a large amount of E according to the prescription (getting a yellowish green). Mixing E with R in the correct proportion ( parts of R and parts of E) yields the color of Q. It is quite remarkable that if BQ is extended to F, and the similar technique is used, the very same color will be used for Q. This is the content of the previous sections. 4. 47 It is now rather clear why the subtraction law (axiom VI) was not true. For the colors “were” the points inside and on the triangle R YB.
3 PHYSICAL MOTIVATIONS IN GEOMETRY In this section are various examples which illustrate how the notion of a center of mass may be used in geometry. EXAMPLE 1 (The Centroid of a Triangle). Consider a triangle ABC and place a mass 1 at each vertex (see Fig. 4). 5 Let M be the center of mass of this system. To find it, replace 1A and 1B with their “equivalent,” 2M1, located at their mid-point. Then replace 2M1 and 1C with their equivalent. The answer is 3M, the point on the median from C and two-thirds of the way from C (see Fig.