GEOMETRICAL PORISMS. 109 



will be in the fame fegment of the circle with the angle BAG j 

 but if BD, CE, lie in contrary diredlions to AB, AC, (fig. 5.) 

 then H will be in that fegment of the circle upon which BAG 

 ftands. 



The point H will be found by the following conflrucflion ; 

 Defcribe a circle through the points A, B, G. Join BC, which 

 divide at G, fo that BG may be to GG in the given ratio of BD 

 to GE, that is of a, to jS, and if the fegments to be cut off are to 

 lie in the fame diredlion with AB, AG, find F the vertex of the 

 fegment upon which the angle BAG ftands, (fig. 4.) ; but if BD, 

 CE are to lie in oppofite directions, (fig. 5.) find F the vertex of 

 the fegment BAG, and in either cafe join FG, which produce to 

 meet the circle in H the point to be found ; that is, if any circle 

 be defcribed through H and A to meet the given lines in D and 

 E, BD is to GE as a to /3. Join HB, HG, HD, HE. The tri- 

 angles BDH, GEH are fimilar, for the angle BDH is equal to 

 CEH, and becaufe the angle BHG is equal to DHE, therefore 

 BHD is equal to GHE ; hence BD is to CE as BH to HG, that 

 is, (becaufe HG bifects the angle BHG), as BG to GG, or as ot 



:tO/3. 



It is evident that the point H may be alfo found, by taking 

 any fegments BD, GE, in the given ratio of ex. to /3, and defcrib- 

 ing a circle through the points D, A, E, to meet the circle BAFG 

 in H the point required. If the given lines be parallel, and the 

 points B, G, alfo the ratio of BD to CE, (fig. 6.) given as before, 

 the indeterminate circle will be changed into a ftraight line paf- 

 fing through a given point H, which will be without the given 

 lines, or between them, according as BD, CE, are to lie in the 

 fame^ or in contrary, direcflions with AB, AG. 



PROP. li. 



