. 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. 3. lines LAM, MBN, NCL touching ^ the circle ABC : there-fore because LM, MN, NL touch the circle ABC in thepoints A, B, C, to which from the centre are drawn KA, KB, c KC, the angles at the points A, B, C are rights angles: andbecause the four angles of the quadrilateral figure AMBK are OF EUCLID.
. 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. 3. lines LAM, MBN, NCL touching ^ the circle ABC : there-fore because LM, MN, NL touch the circle ABC in thepoints A, B, C, to which from the centre are drawn KA, KB, c KC, the angles at the points A, B, C are rights angles: andbecause the four angles of the quadrilateral figure AMBK are OF EUCLID. 105 d equal to four right angles, for it can be divided into two : and that two of them KAM, KBM are right angles, theother two AKB,AMB are equal totwo right angles :but the anglesDEG, DEF arelikewise equaH totwo right angles;therefore the an-gles AKB, AMBare equal to theangles DEG, DEFof which AKB is ^^ B N equal to DEG; wherefore the remaining angle AMB is equalto the remaining angle DEF: in like manner, the angle LNMmay be demonstrated to be equal to DFE; and therefore theremaining angle MLN is equal e to the remaining angle EDF: e the triangle LMN is equiangular to the triangleDEF: and it is described about the circle ABC. Which was tobe PROP. IV. PROB. TO inscribe a circle in a given triangle. SeeN. Let the given triangle be ABC; it is required to inscribe acircle in ABC. Bisect a the angles ABC, BC A by the straight lines BD, CD » 9-1-meeting one another in the point D, from which draw DE, ^, DG perpendiculars to AB, A BC, CA: and because the angleEBD is equal to the angle FBD,for the angle ABC is bisected by BD, and that the right angleBED is equal to the right angleBFD, the two triangles EBD,FBD have two angles of the oneequal to two angles of the other,and the side BD, which is oppo-site to one of the equal angles ineach, is common to both ; there-fore their other sides shall be O
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Keywords: ., bookauthoreuclid, bookcentury1800, booksubje, booksubjectgeometry
