and, conversely, if the rectangle contained by the extremes of three straight lines is equal to the square on the mean the lines are proportional. *Ex. 37. Apply Theor. 10 to shew that, if two chords of a circle intersect either within or without a circle, the rectangle contained by the segments of the one is equal to the rectangle contained by the segments of the other. Ex. 38. If two chords AB, AC drawn from any point A in the circumference of a circle be produced to meet the tangent at the other extremity of the diameter through A in D and E, the triangles AED, ABC are similar. Ex. 39. If ABC be a right-angled triangle, whose right angle B is bisected by BF, cutting the base in F, and meeting the circumference of the circle described about ABC in D, prove that the rectangle contained by BD and BF is equal to twice the area of ABC. THEOR. II. Similar triangles are to one another in the duplicate ratio of their homologous sides. Let ABC, DEF be two similar triangles, having the sides BC, EF homologous: E then shall the triangle ABC be to the triangle DEF in the duplicate ratio of BC to EF. Apply the triangle ABC to the triangle DEF so that B may fall on E and BC along EF, then BA will fall along ED, since the angle ABC is equal to the angle DEF. Hyp. Let A',C' be the positions of A and C in ED, EF or in these sides produced. Join A'F. Because triangles of the same altitude are to one another as their bases, therefore the triangle A'EC' is to the triangle A'EF as EC' is to EF, that is, as BC is to EF. For the same reason, the triangle A'EF is to the triangle DEF as A'E is to DE, that is, as AB : DE; but AB BC:: DE: EF, : therefore AB: DE:: BC: EF, Hyp. Alternando. and therefore the triangle A'EF is to the triangle DEF as BC: EF. Now, the triangle A'EC' has to the triangle DEF the ratio compounded of the ratios of the triangle A'EC' to the triangle A'EF and of the triangle A'EF to the triangle DEF, IV. Def. 7. and each of these ratios has been shewn to be equal to the ratio BC: EF, therefore the triangle A'EC', that is, the triangle ABC, has to the triangle DEF the duplicate ratio of BC to EF. IV. Def. 9. Q.E.D. Ex. 40. ABC is a triangle. AE and BF, intersecting in G, are drawn to bisect the sides BC, AC in E and F. Compare the areas of the triangles AGB, FGE. THEOR. 12. The areas of similar rectilineal figures are to one another in the duplicate ratio of their homologous sides. Let ABCD, HKLM be two similar figures, having the sides AB, HK homologous : then shall the figure ABCD be to the figure HKLM in the duplicate ratio of AB, HK. Let the figures be placed so as to have their homologous sides parallel, and so that one figure is within the other. Let AH, BK, CL, DM be produced to meet in the centre of similarity O, V. 6. then the figures are divided into as many pairs of similar triangles as each figure has sides, V. 6. Cor. and the triangles of each pair are to one another in the duplicate ratio of their bases, V. 11. that is, in the duplicate ratio of homologous sides of the figures, therefore the sum of the triangles that make up the one figure is to the sum of the triangles that make up the other in the duplicate ratio of homologous sides of the figures, IV. 6. that is, the figure ABCD is to the figure HKLM in the duplicate ratio of AB to HK. Q.E.D. COR. Similar rectilineal figures are to one another as the squares described on their homologous sides For the figures and the squares are each to another in the duplicate ratio of the homologous sides. Ex. 41. A square and a hexagon are inscribed in the same circle and equilateral triangles are described on their sides. Find the ratio of a triangle on the side of the square to one on the side of the hexagon. THEOR. 13. If four straight lines are proportional and a pair of similar rectilineal figures are similarly described on the first and second, and also a pair on the third and fourth, these figures are proportional; and conversely, if a rectilineal figure on the first of four straight lines is to the similar and similarly described figure on the second as a rectilineal figure on the third is to the similar and similarly described figure on the fourth, the four straight lines are proportional. Let AB, CD, EF, GH be four straight lines in proportion, and let KAB, LCD be similar and similarly described rectilineal figures on AB, CD, and also MF, NH similar and similarly described figures on EF, GH: K N F G H then shall the figure KAB be to the figure LCD as the figure MF is to the figure NH. Because the ratio AB CD is equal to the ratio EF: GH, therefore the duplicate ratio of AB : CD is equal to the duplicate ratio of EF GH; IV. 14, Part ii. but the figure KAB is to the figure LCD in the duplicate ratio of AB: CD, V. 12. and the figure MF is to the figure NH in the duplicate ratio of EF: GH; V. 12. therefore the figure KAB is to the figure LCD as the figure MF is to the figure NH. Conversely, let a rectilineal figure KAB on AB be to LCD the similar and similarly described figure on CD as a rectilineal figure MF on EF is to the similar and similarly described figure NH on GH: then shall AB be to CD as EF is to GH. The figure KAB is to the figure LCD in the duplicate ratio of AB : CD, V. 12. and the figure MF is to the figure NH in the duplicate ratio of EF: GH; V. 12. |