PROPOSITION XVIII. PROBLEM. Upon a given straight line to describe a rectilinear figure similar and similarly situated to a given rectilinear figure. Η E A B Let AB be the given st. line, and CDEF the given rectil. fig. of four sides. It is required to describe on AB a fig. similar and similarly situated to CDEF. Join DF, and at A and B, make ▲ BAG = ▲ DCF, and LABG= L CDF; then ▲ BAG is equiangular to ▲ DCF. L At G and B, make ▲ BGH= ▲ DFE, and ▲ GBH= ▲ FDE ; L then ▲ GHB is equiangular to ▲ FED. Then AGB= ▲ CFD, and ▲ BGH= 2 DFE, .. LAGH = CFE. So also 4 ABH: L CDE. And we know that ▲ BAG = 4 DCF, ..rectil. fig. ABHG is equiangular to fig. CDEF. Ax. 2. .. GB is to GH as FD is to FE. Also, AG is to GB as CF is to FD. .. AG is to GH as CF is to FE. Similarly, it may shown that GH is to HB as FE is to ED, and that HB is to BA as ED is to DC. .. the rectil. figs. ABHG and CDEF are similar. VI. 4. V. 21. NEXT. Let it be required to describe on AB a fig., similar and similarly situated to the rectil. fig. CDKEF. Join DE, and on AB describe the fig. ABHG, similar and similarly situated to the quadrilateral CDEF. L L At B and H make ▲ HBL= ▲ EDK, and ▲ BHL= ¿ DEK ; then ▲ HLB is equiangular to ▲ EKD. Then the figs. ABHG, CDEF are similar, .. ▲ GHB=L FED; and we have made ▲ BHL= ▲ DEK; L For the same reason, ▲ ABL= 4 CDK. Thus the fig. AGHLB is equiangular to fig. CFEKD. Ax. 2. Again, the figs. AGHB, CFED are similar, .. GH is to HB as FE is to ED: also we know that HB is to HL as ED is to EK, .. GH is to HL as FE is to EK. VI. 4. V. 21. For the same reason, AB is to BL as CD is to DK. And BL is to LH as DK is to KE; ..the five-sided figs. AGHLB, CFEKD are similar. VI. 4. In the same way a fig. of six or more sides may be described, on a given line, similar to a given fig. Q. E. F. PROPOSITION XIX. THEOREM. Similar triangles are to one another in the duplicate ratio of their homologous sides. P A A B C Let ABC, DEF be similar As, having 4 s at A, B, C= 4 s at D, E, F respectively, so that BC and EF are homologous sides. Then must ▲ ABC have to ▲ DEF the duplicate ratio of that which BC has to EF. Suppose DEF to be applied to ▲ ABC, so that E lies on B, ED on BA, and .. EF on BC. Let P and Q be the pts. in BA, BC on which D and F fall. Join AQ. Then A ABC is to ▲ ABQ as BC is to BQ, VI. 1. and ▲ ABQ is to ▲ PBQ as AB is to BP. VI. 1. But AB is to BP as BC is to BQ, VI. 4. V. 5. V. 5. .. ^ ABQ is to ▲ PBQ as BC is to BQ. Hence A ABC is to ▲ ABQ as A ABQ is to ▲ PBQ. .. AABC has to ▲ PBQ the duplicate ratio of A ABC to ▲ ABQ ; .. ▲ ABC has to ▲ PBQ the duplicate ratio VI. Def. 2. V. 5. of BC to BQ. Q. E. D. VI. Def. 2. COR. If MN be a third proportional to BC and EF, BC has to MN the duplicate ratio of BC to EF, and .. BC is to MN as ▲ ABC is to ▲ DEF. Δ Exercises on Proposition XIX. Ex. 1. Prove this Proposition without drawing any line inside either of the triangles. Ex. 2. In the figure, if BC be equal to FD, shew that the triangles will be in the ratio of AB to EF. Ex. 3. Cut off the third part of a triangle by a straight line parallel to one of its sides. Ex. 4. AB, AC are bisected in D and E. Prove that the quadrilateral DBCE is equal to three times the triangle ADE. Ex. 5. ABC is a line passing through the centre of the circle BCD, and AD a tangent to the circle. If CE be drawn parallel to BD, shew that the triangles ACD, ACE are to one another as AC to AB. Ex. 6. A straight line drawn parallel to the diagonal BD of a parallelogram ABCD meets AB, BC, CD, DA, in E, F, G, H. Prove that the triangles AFG, CEH are equal. Ex. 7. If two triangles have an angle equal, and be to each other in the duplicate ratio of adjacent sides, they are similar. Ex. 8. The circle B'C (centre 0) touches the circle ABC internally, and AB'B touches B'C in B'. Shew that if BD be perpendicular to the common diameter, AB, B' divides AB into segments, which are in the duplicate ratio of OC to OD. Ex. 9. From the extremities A, B, of the diameter of a circle, perpendiculars AY, BZ, are let fall on the tangent at any point C. Prove that the areas of the triangles ACY, BCZ are together equal to that of the triangles ACB. Ex. 10. If to the circle, circumscribing the triangle ABC, a tangent at C be drawn, cutting AB produced in D, shew that AD is to DB in the duplicate ratio of AC to CB. Ex. 11. Construct a triangle which shall be to a given triangle in a given ratio. PROPOSITION XX. THEOREM. (Eucl. vi. 21.) Rectilinear figures, which are similar to the same rectilinear figure, are also similar to each other. Let each of the rectilinear figures A and B be similar to the rectilinear figure C. Then must the figure A be similar to the figure B. For. A is similar to C, .. A is equiangular to C, and A and C have their sides about the equals pro portionals. Again, . B is similar to C, VI. Def. 1. .. B is equiangular to C, and B and C have their sides about the equals pro portionals. VI. Def. 1. Hence A and B are each equiangular to C, and have the sides about the equals of each of them and of C proportionals. .. A is equiangular to B, Ax. 1. and A and B have their sides about the equals pro portionals. V. 5. .. the figure A is similar to the figure B. VI. Def. 1. Q. E. D. |