. 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. 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
. 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. 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. 106 THE ELEMENTS Book IV. equal c; wherefore DE is equal to DF: for the same reason,v.^vr-*> DG is equal to DF; therefore the three straight lines DE, DF,c DG are equal to one another, and the circle described from thecentre D, at the distance of any of them, shall pass through theextremities of the other two, and touch the straight lines AB,BC, CA, because the angles at the points E, F, G are right an-gles, and the straight line which is drawn from the extremity ofd 16. 3. a diameter at right angles to it, touchesl the circle: thereforethe straight lines AB, BC, CA do each of them touch the circle,and the circle EFG is inscribed in the triangle ABC. Whichwas to bedone. PROP. V. PROB. See N. TO describe a circle about a given triangle. Let the given triangle be ABC; it is required to describe acircle about Bisect a AB, AC in the points D, E, and from these pointsb draw DF, EF at right angles* to AB, AC; DF, EF produced
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Keywords: ., bookauthoreuclid, bookcentury1800, booksubje, booksubjectgeometry
