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 . tshall be equal to the given triangle ABC, and have one ofits angles equal to Z). Bisecti?aat^:[ AE, and at the pointE, in the straight line EC,make the angle C^i^equaltoi>; [ through A draw AFGparallel to EC, and throughC draw CG parallel toEF. [I. 31. Therefore FECG is a parallelogi-am. [Definition. And, because BE is equal to EC, [Construction* the triangle ABE is equal to the triangle AEC, becausethey are on
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 . tshall be equal to the given triangle ABC, and have one ofits angles equal to Z). Bisecti?aat^:[ AE, and at the pointE, in the straight line EC,make the angle C^i^equaltoi>; [ through A draw AFGparallel to EC, and throughC draw CG parallel toEF. [I. 31. Therefore FECG is a parallelogi-am. [Definition. And, because BE is equal to EC, [Construction* the triangle ABE is equal to the triangle AEC, becausethey are on equal bases BE, EC, and between the sameparallels BC, AG. [I. 38. Therefore the triangle ABC is double of the triangle ^^C. But the parallelogram FECG is also double of the triangleAEC, because they are on the same base EC, and betweenthe same parallels EC, AG. [I. 41. Therefore the parallelogram FECG is equal to the triangleABC; [Axiom 6. and it has one of its angles CEF equal to the given angleD. [Construction. Wherefore a parallelogram FECG has been describedequal to the given triangle ABC, and having one of it»angles CEF equal to the given angle D, BOOK I. 43, 44. 45 PKOPOSITION 43. THEOREM. The complements of the parallelograms which are aboutthe diameter of any parallelogram^ are equal to oneanother. Let ABCD be a parallelogram, of which the diameteris^C; and EH, GF parallelograms about AC, that is,throuiili -which AC passes ; and BK, KD the other paral-lelograms which make up the whole figure ABCD, andwhich are therefore called the complements: the comple-ment BK shall be equal to the complement KD, Because ABCD is aparallelogram, and AC itsdiameter, the triangle ABCis equal to the triangleADC. [I. 34. Again, because AEKH isa parallelogram, and AKits diameter, the triangleAEK is equal to the triangleAHK. [I. 34. For the same reason the triangle KGC is equal to thetriangle KFC. Therefore, because the triangle AEK is equal to the tri-angle AHK, and the triangle KG
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