DE describe a circle meeting AB produced in G. And because AB is bisected in D, and produced to G, (6. 2.)AG.GB+DB2=DG2=DE2. But (47. 1.) DE2 = DB2 + BE, therefore AG. GB + DB2 = DB2 + BE and AG.GB = BE2. Now, BE =C; wherefore the straight line AB is produced to G, so that the rectangle contained by the segments AG, GB of the line produced, is equal to the square of C. Which was to be done. PROP. XXX. PROB. To cut a given straight line in extreme and mean ratio. Let AB be the given straight line; it is required to cut it in extreme. and mean ratio. Upon AB describe (46. 1.) the square BC, and produce CA to D, so that the rectangle CD.DA may be equal to the square CB(29. 6.). Take AE equal to AD, and complete the rectangle DF under DC and AE, or under DC and DA. Then, because D A the rectangle CD.DA is equal to the square line AB is cut in extreme and mean ratio inE Otherwise. F B Let AB be the given straight line; it is required to cut it in extreme and mean ratio. Divide AB in the point C, so that the rectangle contained by AB, BC be equal to the square of AC (11.2.): Then because the rectangle AB BC is equal to the square of AC, as BA to AC, so is AC to CB (17.6.): Therefore AB is cut in extreme and mean ratio in C (def. 3. 6.). Which was to be done. PROP. XXXI. THEOR. In right angled triangles, the rectilineal figure described upon the side opposite to the right angle, is equal to the similar, and similarly described figures upon the sides containing the right angle. Let ABC be a right angled triangle, having the right angle BAC : The rectilineal figure described upon BC is equal to the similar, and similarly described figures upon BA, AC. Draw the perpendicular AD; therefore, because in the right angled triangle ABC, AD is drawn from the right angle at A perpendicular to the base BC, the triangles ABD, ADC are similar to the whole triangle ABC, and to one another (8. 6.), and because the triangle ABC is similar to ADB, as CB to BA, so is BA to BD (4. 6.); and because these three straight lines are proportionals, as the first to the third, so is the figure upon the first to the similar, and similarly described figure upon the second (2. Cor. 20. 6.): Therefore, as CB to BD, so is the figure upon CB to the similar and similarly described figure upon BA: and inversely (B. 5.), as DB to BC, so is the figure upon BA to that upon BC; for the same reason as DC to CB, so is the figure upon CA to that upon CB. Wherefore, as BD and DC together to BC, so are the figures, upon BA and on AC, toge A B D C ther, to the figure upon BC (24.5.); therefore the figures on BA, and on AC, are together equal to that on BC; and they are similar figures. Wherefore, in right angled triangles, &c. Q. E. D. PROP. XXXII. THEOR. If two triangles which have two sides of the one proportional to two sides of the other, be joined at one angle, so as to have their homologous sides parallel to one another; their remaining sides shall be in a straight line. Let ABC, DCE be two triangles which have two sides BA, AC proportional to the two CD, DE, viz. BA to AC, as CD to DE; and let AB be parallel to DC, and AC to DE; BC and CE are in a straight line. Because AB is parallel to DC, and the straight line AC meets them, the alternate angles BAC, ACD are equal (29. 1.); for the same reason, the angle CDE is equal to the angle ACD; wherefore also BAC is equal to CDE: And because the triangles ABC, DCE have AN one angle at A equal to one at D, and the sides about these angles proportionals, viz. BA to AC, as CD to DE, the triangle ABC is equiangular (6. 6.) to DCE : Therefore the angle ABC is equal to the angle DCE: And the angle BAC was proved to be equal to ACD: Therefore the whole angle ACE is equal to the two angles ABC, BAC; add the common angle ACB, then the angles ACE, ACB are equal to the angles ABC, BAC, ACB: But ABC, BAC, ACB are equal to two right angles (32. 1.); therefore also the angles ACE, ACB are equal to two right angles: And since at the point C, in the straight line AC, the two straight lines BC, CE, which are on the opposite sides of it, make the adjacent angles ACE, ACB equal to two right angles; therefore (14. 1.) BC and CE are in a straight line. Wherefore, if two triangles, &c. Q. E. D. PROP. XXXIII. THEOR. In equal circles, angles, whether at the centres or circumferences, have the same ratio which the arches, on which they stand, have to one another: So also have the sectors. Let ABC, DEF be equal circles; and at their centres the angles BGC, EHF, and the angles BAC, EDF at their circumferences; as the arch BC to the arch EF, so is the angle BGC to the angle EHF, and the angle BAC to the angle EDF and also the sector BGC to the sector EHF. Take any number of arches CK, KL, each equal to BC, and any number whatever FM, MN each equal to EF; and join GK, GL, HM, HN. Because the arces BC, CK, KL are all equal, the angles BGC, CGK, KGL are also all equal (27.3.): Therefore, what multiple soever the arch BL is of the arch BC, the same multiple is the angle BGL of the angle BGC: Forthe same reason, whatever multiple the arch EN is of the arch EF, the same multiple is the angle EHN of the angle EHF. But if the arch BL, be equal to the arch EN, the angle BGL is also equal (27. 3.) to the angle EHN; or if the arch BL be greater than EN, likewise the angle BGL is greater than EHN; and if less, less: There being then four magnitudes, the two arches BC, FF, and the two angles BGC, EHF, and of the arch BC, and of the angle BGC, have been taken any equimultiples whatever, viz. the arch BL, and the angle BGL; and of the arch EF, and of the angle EHF, and equimultiples whatever, viz. the arch EN, and the angle EΗΝ : And it has been proved, that if the arch BL be greater than EN, the angle BGL is greater than EHN; and if equal, equal; and if less, less: As therefore, the arch BC to the arch EF, so (def. 5. 5.) is the angle BGC to the angle EHF: But as the angle BGC is to the angle EHF, so is (15. 5.) the angle BAC to the angle EDF, for each is double of each (20.3.): Therefore, as the circumference BC is to EF, so is the angle BGC to the angle EHF, and the angle BAC to the angle Also, as the arch BC to EF, so is the sector BGC to the sector EHF. Join BC, CK, and in the arches BC, CK take any points X, O, and join BX, XC, CO, OK: Then, because in the triangles GBC, GCK, the two sides BG, GC are equal to the two CG, GK, and also contain equal angles; the base BC is equal (4. 1.) to the base CK, and the triangle GBC to the triangle GCK: And because the arch BC is equal to the arch CK, the remaining part of the whole circumference of the circle ABC is equal to the remaining part of the whole circumference of the same circle: Wherefore the angle BXC is equal to the angle COK (27. 3.); and the segment BXC is therefore similar to the segment COK (def. 9. 3.); and they are upon equal straight lines BC, CK: But similar seginents of circles upon equal straight lines are equal (24. 3.) to one another: Therefore the segment BXC is equal to the segment COK: And the triangle BGC is equal to the triangle CGK; therefore the whole, the sector BGC is equal to the whole, the sector CGK. For the same reason, the sector KGL is equal to each of the sectors BGC, CGK; and in the same manner, the sectors EHF, FHM, MHN may be proved equal to one another: Therefore, what multiple soever the arch BL is of the arch BC, the same multiple is the sector BGL of the sector BGC. For the same reason, whatever multiple the arch EN is of EF, the same multiple is the sector EHN of the sector EHF: Now, if the arch BL be equal to EN, the sector BGL is equal to the sector EHN; and if the arch BL be greater than EN, the sector BGL is greater than the sector EHN; and if less, less: Since, then, there are four magnitudes, the two arches BC, EF, and the two sectors BGC, EHF, and of the arch BC, and sector BGC, the arch BL and the sector BGL are any equimultiples whatever; and of the arch EF, and sector EHF, the arch EN and sector EHN, are any equimultiples whatever; and it has been proved, that if the arch BL be greater than EN, the sector BGL is greater than the sector EHN; if equal, equal; and if less, less; therefore (def. 5. 5.), as the arch BC is to the arch EF, so is the sector BGC to the sector EHF. Wherefore, in equal circles, &c, Q. E. D. PROP. B. THEOR. If an angle of a triangle be bisected by a straight line, which likewise cuts the base; the rectangle contained by the sides of the triangle is equal to the rectangle contained by the segments of the base, together with the square of the straight line bisecting the angle. Let ABC be a triangle, and let the angle BAC be bisected by the straight line AD; the rectangle BA. AC is equal to the rectangle BD.DC, together with the square of AD. Describe the circle (5. 4.) ACB about the triangle, and produce AD to the circumference in E, and join EC. Then, because the angle BAD is equal to the angle CAE, and the angle ABD to the angle B A C (21. 3.) AEC, for they are in the same segment; the triangles ABD, AEC are equiangular to one another: Therefore BA: AD :: EA: (4.6.) AC, and consequently, BA.AC = (16. 6.) AD.AE = ED.DA (3. 2.) + DA2. But ED.DA = D BD.DC, therefore BA.AC = BD.DC + E DA3. Wherefore, if an angle, &c. Q. E. D. |