120 G eomet:rical porisms. 



PROP. VIII. PORISM, Fig. 13. PI. IL 



Let CA, CB, AB be three llraight lines given by poGtion, a 

 point H may be found, fuch, that if through H, and B, C, 

 any two of the interfe(5lions of thefe lines, there be defcribed 

 circles HBEF, HCDE, to meet each other at E, a point in 

 BC, and the remaining Hnes at D and F. If DE, £F, DF 

 be joined, the triangle DEF fhall be fimilar to a given triangle 

 def^ and fliall have its angles upon the given lines in a given 

 order. 



Because circles are defcribed through C, B, and meeting 

 each other at E, a point in CB, therefore their other inter feifliou 

 H, the remaining angle A, and the points D, F, are in a circle. 

 (Lemma.) Let a circle be defcribed through H, C, A, 'to meet 

 CB in G, and another through H, B, G, to meet AB in K. 

 Join HA, HG, HK, alfo HD, HE, HF. The angles ADH, 

 GEH, KFH, are equal to one another, and the angles CAH, 

 CGH, BKH are equal, therefore HAD, HGE, HKF are equal, 

 and the triangles HAD, HGE, HKF are fimilar ; therefore DH 

 is to HE as AH to HG, and EH is to HF as GH to HK ; now, 

 the angles DHE, EHF are equal to DCE, EBF, that is, to 

 AHG, GHK ; hence the qiiadrilateral HDEF is fimilar to 

 HAGK, and the triangle DEF is fimilar to AGK ; now, the an- 

 gles EDF and DEF are given by hypothefis, therefore GAK 

 and AGK are given ; but A is a given point, and AK is given 

 by pofition, therefore AG and the point G are given ; therefore 

 GK and the point K are alfo given, and H, the interfed:ion of 

 the given circles GAC, GBK, will be given, which was to be 

 found. 



Construction. Take a given point, which, to render the 

 conflru(flion more fimple, may be at A, one of the interfedlions 



of 



