The elements of Euclid for the use of schools and colleges : comprising the first two books and portions of the eleventh and twelfth books; with notes and exercises . be on the point E, and the stniightline BC on the straight line EF. the point C will also coin-cide ^\-ith the point F, because BC is equal to EF. [, BC coinciding with EF, BA and AC will coin-cide with ED and DF. For if the base BC coincides with the base EF, but thesides BA, CA do not coincide with the sides ED, FD, buthave a different situation as EG, EG ; then on the samebase and on the same side of it there will
The elements of Euclid for the use of schools and colleges : comprising the first two books and portions of the eleventh and twelfth books; with notes and exercises . be on the point E, and the stniightline BC on the straight line EF. the point C will also coin-cide ^\-ith the point F, because BC is equal to EF. [, BC coinciding with EF, BA and AC will coin-cide with ED and DF. For if the base BC coincides with the base EF, but thesides BA, CA do not coincide with the sides ED, FD, buthave a different situation as EG, EG ; then on the samebase and on the same side of it there will be two triangleshaving their sides which are terminated at one extremityof the base equal to one another, and likewise their sideswhich are tenuinated at the other this is impossible. [I. 7. Therefore since the base BC coincides with the base EF,the sides BA, AC must coincide with the sides ED, also the angle BAC coincides with the angleEDF, and is equal to it. {Axiom 8. Wherefore, if two triangles &c. PEOPOSITION 9, PROBLEM. To hisect a giten rectilineal angle, that is to dicidc itinto two equal angles. BOOK I. 9, 10. 15. Let BAChe the given rectilinealangle : it is required to bisect it. Take any point D in AB, andfrom AC cut off AE equal toAD; [I. 3. join DE, and on DE, on the sideremote from A, describe the equi-lateral triangle DBF. [I. 1. Join AF, The straight line AF shall bisect the angle BAG. Because ^Z> is equal to ^^, [Construction. and ^i^ is common to the two triangles DAF, EAF,the two sides DA, AF are equal to the two sides EA, AF^each to each; and the base DF is equal to the base EF; [Definition the angle I)AF is equal to the angle EAF. [I. 8. Wherefore the given rectilineal angle BAG U bisectedby the straight line AF. PROPOSITION 10. PROBLEM,To bisect a given finite straight line, that is to divide iti?ito two equal parts. Let AB he the given straightline : it is required to divide it intotwo equal parts.
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