GEOMEtP^ICAL PORISkS. riy 



Through P and B, the interfe^fi:ion of any two of the given 

 lines, let a circle be defcribed to touch one of them at B, and 

 cut the other at N, the line BN will be given, and the ratios of 

 GH, HK, KL, the fame with the given ratios of BC, CE, EN 

 to one another. 



The fynthetical demonftration follows readily from the ana- 

 lyfis, and for the fake of brevity is here omitted. 



Cor. I. The lines PG, PH, PK, PL, contain given angles^ 

 and have to each the given ratios of PB, PC, PE, PN. 



Cor. 2. The line GL cuts off from the given lines, fegments 

 BG, CH, EK, NL, adjacent to given points, and having to each 

 other the given ratios of PB, PC, PE, PN. For the points P, A, 

 G, H, being in a circle, the angle PGB is equal to PHC ; and 

 fince P, F, H, K, are in a circle, the angle PHC is equal to PKE, 

 which in like manner vvdll be found equal to PLN. Now, the 

 angles PBA, PCF, PEF, PNB are equal among themfelves, there- 

 fore their fupplements PEG,. PCH, PEK, PNL are equal, and 

 'the triangles PBG, PCH, PEK, PNL are fimilar, therefore BG, 

 CH, EK, NL are proportional to the given lines BP, CP, EP^ 

 NP. 



PROP. Vn. THEOREM, Fig. 12. PI. IIL 



Let PGAB, PFAC, PEAD, Uc. be any number of given 

 circles, each of which pafTes through the fame two points 

 A^ P ; from A, either of thefe points let a ftraight line, given 

 by pofition, be drawn, meeting the circles at B, C, D, tffc. 

 and another meeting them at E, F, G, life. Let ftraight 

 hnes GB, FC, ED, isc. be drawn, joining thefe points, fo 

 as to form, with the lines paffing through A, triangles GAB, 

 FAC, EAD, i^c. in each of the circles. If, through P, the 

 common interfecftion of the circles, and Q^ the interfedion 



o± 



