Articoli correlati a Elements of geometry Volume 1
Sinossi
This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1896 Excerpt: ...Angles formed by homologous lines are homologous. PROPOSITION XI. THEOREM 298. Two polygons radially situated arc similar and their ratio of similitude is equal to the determining ratio. Given--the polygons ABODE and A'B'C'D'E' radially situated, O being the ray centre. To Prove--they are similar, and that the determining ratio is their ratio of similitude. AB is parallel to A'B', BC to B'C, etc. § 273 If a straight line divide two sides of a triangle proportionally, it is parallel to the third side. Hence angle ABC=A'B'C angle BCD=B'C'D', etc. § 51 Having their sides respectively parallel and in the same right-and-left order. Again, triangle OAB is similar to OA'B', OBC to OB'C. etc. § 285 _,. AB (0B BC (0C „ Therefore = ) = = ( ) = etc. § 274 A'B' OB'J B'C OC'J 8.,,, AB BC Whence = = etc. Ax 1 A'B' B'C Since the polygons have their angles respectively equal and their homologous sides proportional, they are similar. §274 AB Also, their ratio of similitude = determining ratio 0B_ A'B' OB' Q-E-D 299. De/.--The ray centre is also called the centre of similitude. PROPOSITION XII. THEOREM 300. A ny two similar polygons can be radially placed, the determining ratio being equal to the ratio of similitude. Given the similar polygons P and p. To Prove--that they can be radially placed, the determining ratio being the ratio of similitude. With any point 0 as ray centre form a polygon/' radially situated with regard to P, having the determining ratio Oa' ab equal to the ratio of similitude of p and P. OA 4 AB F Then /' and P will be similar, the ratio of similitude being a'b' Oa' Q Q---= § 298 AB OA But p and P are given similar, and their ratio of similitude ab "AB Therefore p' and p are similar. § 294 That is, the ratio...
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