By George David Birkhoff, Ralph Beatley
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Extra resources for Basic Geometry - Manual for Teachers
Note that the preceding exercises contain geometric configurations that resemble those needed in the proof of this theorem. Page 80, lines 13 and 15. Read "sum of" both times, and "difference between" both times, to cover both cases shown in Fig. 15. Pages 80-82: Exercises. 1. Angles ABC, ABF, EPG, EFB can vary. Also angles DCB, DCG, HGF, HOC. 4. Five distinct cross-braces can be made. for rigidity. Only two of these are needed These two can be chosen in ten different ways. 6. The fact that these Ideas on equidistance are usually presented in e locus theorem receives recognition In Locus Theorem 4 in Chapter 9.
A responding sides of two triangles" to "sides opposite equal angles" in these triangles, because it Is often possible to consider two triangles - 49 - that are related like those shown in Fig. A on page 49 of this manual, in which the vertices, sides, and angles correspond in pairs despite the absence of equality. In short, Euclidean geometry can be regarded as a special case of projective geometry. The term "correspond" defies defi- nition for beginners, but everyone knows what it means. The word "proportion" is explained here but is not precisely defined.
7-10, so the proofs of Us. 11 and 12 are merely obvious extensions of the preceding exercises. 16. Notice the gradual build-up of the perpendicular-bisector locus in this book as revealed on pages 63, 81, 88-89, 250-251. Notice that all the ideas essential to a locus are given relatively early, but that the use of the word "locus" itself In this connection is withheld until page 250. - 53 - 17. By no means tell the pupil which three of the four small triangles be in to work with. Let him discover this for himself.