. The Elements of Euclid : viz. the first six books, together with the eleventh and twelfth : the errors, by which Theon, or others, have long ago vitiated these books, are corrected, and some of Euclid's demonstrations are restored : also, the book of Euclid's Data, in like manner corrected. B C Wherefore, equal triangles, &c. PROP. XL. THEOR. EQUAL triangles upon equal bases, in the samestraight line, and towards the same parts, are betweenthe same parallels. Let the equal triangles ABC, DEF be upon equal bases BC,EF, in the same straightline BF, and towards thesame parts; they are be-tween
. The Elements of Euclid : viz. the first six books, together with the eleventh and twelfth : the errors, by which Theon, or others, have long ago vitiated these books, are corrected, and some of Euclid's demonstrations are restored : also, the book of Euclid's Data, in like manner corrected. B C Wherefore, equal triangles, &c. PROP. XL. THEOR. EQUAL triangles upon equal bases, in the samestraight line, and towards the same parts, are betweenthe same parallels. Let the equal triangles ABC, DEF be upon equal bases BC,EF, in the same straightline BF, and towards thesame parts; they are be-tween the same parallels. Join AD; AD is paral-lel to BC : for, if it is not,through A draw ^ AG pa-rallel to BF, and join GF: „ r v t?the triangle ABC is equal »> ^ *- £. ^ b ,to the triangle GEF, because they are upon equal bases BC, EF,and between the same parallels BF, AG : but the triangle ABCis equal to the triangle DEF ; therefore also the triangle DEFis equal to the triangle GEF, the greater to the less, which isimpossible : therefore AG is not parallel to BF : and in the samemanner it can be demonstrated that there is no other parallel toit but AD; AD is therefore parallel to BF. Wherefore, equaltriangles, &c. Q. E. a PROP. XLI. THEOR. IF a parallelogram and triangle be upon the samebase, and between the same parallels; the parallelo-gram shall be double of the triangle. 44 THE ELEMENTS Book I. Let the parallelogram ABCD and the triangle EBC be «—V— the same base BC, and between the same parallels BC, AEparallelogram ABCD is double of thetriangle EBC. Join AC; then the triangle ABC a is equal * to the triangle EBC, becausethey are upon the same base BC, andbetween the same parallels BC, AE. b 34. 1. But the parallelogram ABCD is double^of the triangle ABC, because the diame-ter AC divides it into two equal parts;wherefore ABCD is also double of thetriangle EBC. Therefore, if a parallelo-gram, Sec. Q. E. D. upon;the
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