In bd take any point f, and from ae the greater, cut off ag equal (i. 3) to a af, the less, and join fc, gb. Because af is equal to ag, and ab to ac, the two sides fa, ac are equal to the two ga, ab, each to each; and they contain the angle fag common to the two triangles afc, agb; there f d a b C g e fore the base fc is equal (i. 4) to the base gb, and the triangle afc to the triangle agb; and the remaining angles of the one are equal (i. 4) to the remaining angles of the other, each to each, to which the equal sides are opposite; viz. the angle acf to the angle abg, and the angle afc to the angle agb: and because the whole af is equal to the whole ag, of which the parts ab, ac, are equal; the remainder bf shall be equal (3 ax.) to the remainder cg; and fc was proved to be equal to gb; therefore the two sides bf, fc are equal to the two cg, gb, each to each; and the angle bfc is equal to the angle cgb, and the base bc is common to the two triangles bfc, cgb; wherefore the triangles are equal (i. 4), and their remaining angles, each to each, to which the equal sides are opposite; therefore the angle fbc is equal to the angle gcb, and the angle bcf to the angle cbg: and since it has been demonstrated, that the whole angle abg is equal to the whole acf, the parts of which, the angles cbg, bcf are also equal; the remaining angle abc is therefore equal to the remaining angle acb, which are the angles at the base of the triangle abc: and it has also been proved that the angle fbc is equal to the angle gcb, which are the angles upon the other side of the base. Therefore the angles at the base, &c. Q. E. D. COROLLARY. Hence every equilateral triangle is also equiangular. If two angles of a triangle be equal to one another, the sides which subtend, or are opposite to, the equal angles, shall also be equal to one another. LET abc be a triangle having the angle abc equal to the angle acb; the side ab is also equal to the side ac. For, if ab be not equal to ac, one of them is greater than the other: let ab be the greater; and from it cut (i. 3) off db equal to ac, the less, and join dc; therefore, because in the triangles dbc, acb, db is equal to ac, and be common to both, the two sides, db, bc are equal to the two ac, cb, each to each; and the angle dbc is equal to the angle acb; therefore the base dc is equal to the base ab, and the triangle dbc is equal to the triangle (i. 4) acb, the less to the greater; which is absurd. Therefore ab is not unequal to ac, that is, it is equal to it. Wherefore, if two angles, &c. Q. E. D. b d Cor. Hence every equiangular triangle is also equilateral. a PROPOSITION VII. - THEOREM. Upon the same base, and on the same side of it, there cannot be two triangles that have their sides which are terminated in one extremity of the base equal to one another, and likewise those which are terminated in the other extremity. d IF it be possible, let there be two triangles acb, ad b, upon the same base a b, and upon the same side of it, which have their sides ca, da terminated in the extremity a of the base equal to one another, and likewise their sides cb, db, that are terminated in b. a Join cd; then, in the case in which the vertex of each of the triangles is without the other triangle, because acis equal to ad, the angle acd is equal (i. 5) to the angle adc: but b the angle acd is greater than the angle bcd; therefore the angle adc is greater also than bcd; much greater then is the angle bdc than the angle bcd. Again, because cb is equal to db, the angle bdc is equal (i. 5) to the angle bed; but it has been demonstrated to be greater than it; which is impossible. e f C d But if one of the vertices, as d, be within the other triangle acb; produce ac, ad to e, f; therefore, because ac is equal to ad in the triangle acd, the angles ecd, fdc upon the other side of the base cd are equal (i. 5) to one another: but the angle ecd is greater than the angle bcd: wherefore the angle fdc is likewise greater thanbcd; much greater then is the angle bdc than the angle bcd. Again, because cb is equal to db, the angle bdc is equal (i. 5) to the angle bcd; but bdc has been proved to be greater than the same bcd; which is im a b possible. The case in which the vertex of one triangle is upon a side of the other, needs no demonstration. Therefore, upon the same base, and on the same side of it, &c. Q. E. D. PROPOSITION VIII. - THEOREM. If two triangles have two sides of the one equal to two sides of the other, each to each, and have likewise their bases equal; the angle which is contained by the two sides of the one shall be equal to the angle contained by the two sides equal to them, of the other. LET abc, def be two triangles, having the two sides ab, ac, equal to the two sides de, df, each to each, viz. ab to de, and acto df; and also the base be equal to the dg Ce f For, if the triangle abc be applied to def, so that the point b be on e, and the straight line bc upon ef; the point c shall also coincide with the point f, beb cause bc is equal to ef. Therefore bc coinciding with ef; ba and ac shall coincide with ed and df; for, if the base be coincides with the base ef, but the sides ba, ca do not coincide with the sides ed, fd, but have a different situation, as eg, fg, then upon the same base ef, and upon the same side of it, there can be two triangles that have their sides which are terminated in one extremity of the base equal to one another, and likewise their sides terminated in the other extremity; but this is impossible (i. 7); therefore, if the base bc coincides with the base ef, the sides ba, ac, cannot but coincide with the sides ed, df; wherefore likewise the angle bac coincides with the angle ed f, and is equal (8ax.) to it. Therefore if two triangles, &c. Q.E.D. PROPOSITION IX. - PROBLEM. To bisect a given rectilineal angle, that is, to divide it into two equal angles. LET bac be the given rectilineal angle, it is required to bisect it. Take any point d in ab, and from ac cut (i. 3) off ae equal to ad; join de, and upon it describe (i. 1) an equilateral triangle def; then join af; the straight line af bisects the angle bac. a Because ad is equal to ae, and af is common to the two triangles daf, eaf; the two sides da, af, are equal to the two sides ea, af, each to each; and the base dfis equal to the base ef; therefore the angle daf is equal (i. 8) to the angle eaf; wherefore the given rectilineal angle bacis bisected b by the straight line af. Which was to be done. PROPOSITION X. - PROBLEM. To bisect a given finite straight line, that is, to divide it into two equal parts. LET ab be the given straight line, it is required to divide it into two equal parts. Describe (i. 1) upon ab an equilateral triangle abc, and bisect (i. 9) the angle ac b by the straight linecd. abis cut into two equal parts in the point d. Because ac is equal to cb, and cd common to the two triangles acd, bcd; the two sides a c, cd are equal to bc, cd, each to each; and the angle acd is equal to the angle bcd; therefore the base ad is equal to the base (i. 4) db, and the bstraight line abis divided into two equal parts in the point d. Which was to be done. PROPOSITION XI. - PROBLEM. To draw a straight line at right angles to a given straight line, from a given point in the same. LET ab be a given straight line, and ca point given in it; it is required to draw a straight line from the point c at right angles to a b. Take any point din ac, and make (i. 3) ce equal to cd, and upon de describe (í. 1) the equilateral triangle dfe, and join fc, the straight line fc drawn from the given point c is at right angles to the given straight line a b. f a d C e Because dc is equal to ce, and fc common to the two triangles dcf, ecf; the two sides dc, cf, are equal to the two ec, cf, each to each; and the base bd is equal to the base ef; therefore the angle dcfis equal (i. 8) to the angle ecf; and they are adjacent angles. But, when the adjacent angles which one straight line makes with another straight line are equal to one another, each of them is called a right angle (10 def.); therefore each of the angles dcf, ecf, is a right angle. Wherefore, from the given point c, in the given straight line a b, fc has been drawn at right angles to ab. Which was to be done. COR. By help of this problem, it may be demonstrated, that two straight lines cannot have a common segment. a e If it be possible, let the two straight lines abc, abd have the segment ab common to both of them. From the point b draw be at right angles to ab; and because abc is a straight line, the angle cbe is equal (10 def.) to the angle eba; in the same manner, because abd is a straight line, the angle dbe is equal to the angle eba; wherefore the angle dbe is equal to the angle cbe, the less to the greater, which is impossible; therefore b d two straight lines cannot have a common segment. PROPOSITION XII. - PROBLEM. To draw a straight line perpendicular to a given straight line of an unlimited length, from a given point without it. LET a b be the given straight line, which may be produced to any length both ways, and let c be a point without it. It is required to draw a straight line perpendicular to ab from the point c. Take any point d upon the other side of ab, and from the centre c, at the distance cd, describe (3 e post.) the circle egf meeting a bin f, g; and bisect (i. 10) fg in h, and join h cf,ch, cg; the straight line ch, drawn from the d given point c, is perpendicular to the given straight line a b. Because fh is equal to hg, and he common to the two triangles fhc, ghc, the two sides fh, hcare equal to the two gh, hc, each to each; and the base cfis equal (15 def.) to the base cg; therefore the angle chf is equal (i. 8) to the angle chg; and they are adjacent angles; but when (10 def.) a straight line standing on another straight line makes the adjacent angles equal to one another, each of them is a right angle; and the straight line which stands upon the other is called a perpendicular to it; therefore from the given point ca perpendicular ch has been drawn to the given straight line a b. Which was to be done. 4 PROPOSITION XIII. - THEOREM. The angles which one straight line makes with another upon one side of it are either two right angles, or are together equal to two right angles. LET the straight line ab make with cd, upon one side of it, the angles cba, abd: these are either two right angles, or are together equal to two right angles. For if the angle cba be equal to abd, each of them is a right angle (10 def.); but if not, from the point b draw be at right angles (i. 11) to cd; therefore the angles cbe, ebd are two right angles (10 def.); and because cbe is equal to the two angles cba, abe together, add the angle ebd |