and Loci of Apollonius, ^c. 83 



for QD or NN equidistant from C and parallel to CG ; and 

 .•. also two points Q, one in eaeli line NN. 



PORISM. 



Given the lines MM, NN, and the points P and Q therein 

 — one hi each line ; given also a point C and a ratio - : innu- 

 merable 2)oints B and corresponding straight lines LL parallel 

 to any given direction can be found, such that I being any jwint 

 in LL and E, F, the points in ivhich IB and IC cut MM and 

 NN, vje shall have PE : QF : : m : u. 



For QCri is known in position, and CG parallel to NN is 

 known in position, and CD parallel to the given direction is 

 known in position, and DA making the angle DA right to C 

 = NN right to MM is known in position ; and the circle QPE, 

 is known. And since PU : : QD : : m : n, the point U is 

 known, and hence UB parallel to CD is known. And R 

 being any point in the circle which passes through the inter- 

 section of MM and NN, and through the points P and Q, 

 the lines HP and RQ will cut UB and DA in B and A, 

 making the triangles DAQ, UBP similar ; therefore drawing 

 BG parallel to MM to cut CG in G, we have the triangles 

 BGII similar to QDA, and therefore LL through G parallel 

 to CD is a required line LL, and B is its corresponding point 

 (the locus of B is evidently a known straight line UB). 



If -^ be not restricted as to sign, then evidently the entire 

 locus of B is two known parallels equidistant from P, &c., &c. 



PORISM. 



Given tvm points B, C, and two lines MM, NN, in jwsition, 

 and a point P in MM ; a point Q can be found in NN, such 

 that straight lines BI, CI, to any point I in a determinable 

 straight line LL shall cut MM and NN in E and F, so that 

 PE shall have to QF the given ratio of m. to n. 



For wc have BG and 'CG parallel to MM and NN ; and 

 therefore as the triangle CDA is similar to BGC, the locus of 

 A is a known straight line KK. And as QA lies between 

 NN, KK, making the angle QA right to N = PB right to 

 M, and that it has to PB the given ratio n : m, .-. QA is 

 evidently known in magnitude and position ; hence, the line 

 CD (making the angle CD right to A = BG right to C) is 

 known, and .•. LL through G parallel to CD. 



G 2 



