io8 GEOMETRICAL PORISMS. 



amples of their application are added, feme of which are pro- 

 blems that have been long known, and others are new ; but the 

 conftrudlions of the former, it is believed, differ from any hi- 

 therto pnblilhed. Although there are feveral of thefe examples-, 

 in appearance, little related to each other, yet their folntions are 

 effedled by the fame general principle, which is alfo the founda^- 

 tion of all the porifms. 



" PROP. I. PORISM, Fig. 4, 5. PI. I. 



Let AB, AC, be two ftraight lines given by polition, let B, C, 

 be given points in thefe lines, a point H may be found, 

 fuch, that any ciicle whatfoever pafling through A, the in- 

 terfecflion of the given lines, and H the point which may 

 be found, fhall cut off from the given lines fegments BD, 

 CE, adjacent to the given points, and having to each, other 

 the given ratio of a to /3. 



Suppose the porifm to be true, and that the point H is found.- 

 If a circle be defcribed through H, A, and B one of the given 

 points, it muft alfo, pafs through C the other given point, that 

 the propofition may be univerfally true. Therefore H is in the 

 circumference of a given circle. Join BH, CPI, DH, EH. The 

 angle DHE is equal to DAE, that is, to BHC, (fig. 4.) or DHE 

 is the fupplement of DAE, (fig. 5.) and therefore equal to BHC ; 

 hence BHD is equal to CHE, but BDH is equal to CEH, there- 

 fore the triangles BDH, CEH, are equiangular, and BH is to 

 HC as BD to CE, that is by hypothefis in the given ratio of a to 

 jS ; therefore if BC be joined, the triangle BHC is given in fpe- 

 cies, and BC being given, BH and HC are given ; therefore the 

 point H is given, which was to be found. 



If the fegments BD, CE, cut off from the given lines, lie in 

 the fame direction with refpedl to AB, AC, (fig. 4.) the point H 



will 



