. 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. PROP. XXL THEOR. THE angles in the same segment of a circle are Sec nequal to one another. Let ABCD be a circle, and BAD,BED angles in the same segmentBAED: the angles BAD, BED areequal to one another. Take F the centre of the circleABCD: and, first, let the segmentBAED be greater than a semicircle,and jo
. 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. PROP. XXL THEOR. THE angles in the same segment of a circle are Sec nequal to one another. Let ABCD be a circle, and BAD,BED angles in the same segmentBAED: the angles BAD, BED areequal to one another. Take F the centre of the circleABCD: and, first, let the segmentBAED be greater than a semicircle,and join BF, FD : and because theangle BED is at the centre, and theangle BAD at the circumference,and that they have the same part of. 86 THE ELEMENTS Booklll. the circumference, viz. BCD, for their base; therefore the an-*>—-v—^ gle BFB is double » of the angle BAD: for the same reason,a the angle BFD is double of the angle BED: therefore the angle BAD is equal to the angle BED. But, if the segment BAED be not greater than a semicircle, let BAD, BED be angles in it; these also are equal to one another: draw AF to the centre, and produce it to C, and join CE : therefore the seg-ment BADC is greater than a semi-circle ; and the angles in it BAG, BEC are equal, by the first case: for the same reason, because CBED is greater than a semicircle, the angles CAD, CED are equal: therefore the whole angle BAD is equal to the whole angle BED. Wherefore, the angles in the same segment, &:c. Q. E. D.
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