. 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. inscribe an equilateral and equiangular penta-gon in a given circle. Let ABCDE be the given circle ; it is required to inscribe anequilateral and equiangular pentagon in the circle ABCDE. Describe* an isosceles triangle FGH, having each of the a at G, H, double of the angle at F ; and in the ci
. 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. inscribe an equilateral and equiangular penta-gon in a given circle. Let ABCDE be the given circle ; it is required to inscribe anequilateral and equiangular pentagon in the circle ABCDE. Describe* an isosceles triangle FGH, having each of the a at G, H, double of the angle at F ; and in the circleABCDE inscribe ^ the triangle ACD equiangular to the tri- b 2. 4angle FGH, so that the angle A CAD be equal to the angleat F, and each of the anglesACD, CDA equal to theangle at G or H ; whereforeeach of the angles ACD,CDA is double of the angle CAD. Bisect c the angles / \ \\ / ^<C^ \ // c , CDA by the straightlines CE, DB ; and join AB,BC, DE, EA. ABCDE isthe pentagon required. Because each of the angles ACD, CDA is double of CAD,and are bisected by the straight lines CE, DB, the five anglesDAC, ACE, ECD, CDB, BDA are equal to one another; butequal angles stand upon equal <i circumferences; therefore (he d 26. circumferences AB, BC, CD, DE, EA are eoual to one. il2 THE ELEMENTS Book IV. another: and ecual circumferences are subtended by equal«^-—v^^ straight lines ; therefore the five straight lines AB, BC, CD,c 29. 3. DE, EA are equal to one another. Wherefore the pentagonABCDE is equilateral. It is also equiangular; because thecircumference AB is equal to the circumference DE : if to eachbe added BCD, the whole ABCD is equal to the whole EDCB :and the angle AED stands on the circumference ABCD, andthe angle BAE on the circumference EDCB; therefore the£ angle BAE is equal f to the angle AED: for the same reason,each of the angles ABC, BCD, CDE is equal to the angle BAE,or AED : therefore the pentagon ABCDE is equiangular; andit has been shown
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Keywords: ., bookauthoreuclid, bookcentury1800, booksubje, booksubjectgeometry
