Elements of geometry and trigonometry . PROPOSITION XV. THEOREM. If two straight lines he cut by three parallel planes, theij will bedivided proportionally. Suppose the line AB to meetthe parallel planes MN, PQ,RS, at the points A, E, B ; andthe line CD to meet the sameplanes at the points C, F, D :we are now to show thatAE : EB : : CF : AD meeting the planePQ in G, and draw AC, EG,GF, BD ; the intersections EG,BD, of the parallel planes PQ,RS, by the plane ABD, areparallel (Prop. X.) ; therefore AE : EB : : AG : GD ;in like manner, the intersections AC, GF, being parallel, AG : GD : :
Elements of geometry and trigonometry . PROPOSITION XV. THEOREM. If two straight lines he cut by three parallel planes, theij will bedivided proportionally. Suppose the line AB to meetthe parallel planes MN, PQ,RS, at the points A, E, B ; andthe line CD to meet the sameplanes at the points C, F, D :we are now to show thatAE : EB : : CF : AD meeting the planePQ in G, and draw AC, EG,GF, BD ; the intersections EG,BD, of the parallel planes PQ,RS, by the plane ABD, areparallel (Prop. X.) ; therefore AE : EB : : AG : GD ;in like manner, the intersections AC, GF, being parallel, AG : GD : : CF : FD ;the ratio AG : GD is the same in both: henceAE : EB : : CF : PROPOSITION XVI. THEOREM. If a line is perpendicular to a pla7ie, every plane passed throughthe perpendicular, will also be pe?pendiculai to the plane. BOOK XL 137 Let AP be perpendicular to theplane NM ; tlien will every planepassing througli AP be perpendicu-lar to i\M. Let BC be the intersection of theplanes AB, MN ; in the plane MN,draw DE perpendicular to BP: thenthe line AP, oeing perpendicular tothe plane MN, will be perpendicu-lar to each of the two straifjht lines M B
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Keywords: ., bo, bookcentury1800, booksubjectgeometry, booksubjecttrigonometry
