Plane and solid geometry . tlie endsof a given line is the perpendicular bisector of that line. FP A^ t- O ^ j^::: C Fig. 1. B R ^N ^B C Fig. 2. Given line AB and its X bisector, CF. To prove CF the locus of all points equidistant from A and B. First Method (Fig. 1)Argument 1. Let P be any jjoint in CF. Then F is equi- distant from A and B, every pointin Ci^ satisfies the prescribed condition. 2. Let Q be any point not in CF. Then Q is unequally distant from A and B, no point outside of CF satisfiesthe prescribed condition. 3. .-. CF is the locus of all points equi- distant from A and
Plane and solid geometry . tlie endsof a given line is the perpendicular bisector of that line. FP A^ t- O ^ j^::: C Fig. 1. B R ^N ^B C Fig. 2. Given line AB and its X bisector, CF. To prove CF the locus of all points equidistant from A and B. First Method (Fig. 1)Argument 1. Let P be any jjoint in CF. Then F is equi- distant from A and B, every pointin Ci^ satisfies the prescribed condition. 2. Let Q be any point not in CF. Then Q is unequally distant from A and B, no point outside of CF satisfiesthe prescribed condition. 3. .-. CF is the locus of all points equi- distant from A and B. Second Method (Fig. 2)Argument 1. Same as Arg. 1, above. 2. Let R be any point such that RA =BB, Then R lies in CF, every pointwhich satisfies the prescribed condi-tion lies in CF. 3. Same as Arg. 3^ above. Reasons1. § 134. 2. § 110. 3. § 130. Reasons 2 § 139. 298 PLANE GEOMKTRY (b) TIxe locus of the mid-points of all cliords of a circleparallel to a given line is the dicnneter perpendicular tothe -B Given circle 0, line AB^ and diameter Its _L AB,To prove ES the locus of the mid-points of all chords ofcircle 0 that are II AB, Argument 1. Let P be any point in diameter /?s. Through P draw FG II AB. 2. Now ES ± AB. 3. .-. i?5 ± FG. 4. .. P is the mid-point of FG^ a chord II AB. 5. Let Q be any point not in diameter RS. Through Q draw UK II AB^ intersectingas in T. 6. Then RS ± HK. 7o .-. r is the mid-point of HK^ Q is notthe mid-point of HK, a chord II AB. 8. 0*. RS is the locus of the mid-points ofall chords of circle 0 that are II AB. Reasons 1. § 179. 2. By hyp. 3. §193. 4. § 302. 5. § 179. 6. i. § 193.§ ;!02. 8. § 130. SOLID GEOMETRY BOOK VI LINES, PLANES, AND ANGLES IN SPACE 602. Def. Solid geometry or the geometry of space treats offigures whose parts are not all in the same plane. (For defini-tion of plane or plane surface, see § 34.) 603. From the definition of a plane it follows that: (a) If two points of a straight line
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectgeometr, bookyear1912
