GEOMETRICAL PORISMS. iii 



PROP. IlL PORISM, Fig. 8. PI. IL 



Let AF, AG be two ftraight lines given by pofition, a point 

 H may be found, fuch, that if any circle be defcribed 

 through it, and A the interfedlion of the given lines, to 

 meet them in D and E, the difference between AD and AE 

 fliall be equal to a given line N. 



The analyfis of this proposition will differ in nothing material 

 from the lift, and the point required may be found thus : Take 

 B and C, two given points, fo that the difference between BA 

 and AC may be equal to N. Through the points A, B, C, de- 

 fcribe a circle. Draw AK bifedling the angle contained by FA 

 one of the given lines, and AL the other line produced at their 

 ihterfedlion, and AK will meet the circle ABC in H the point 

 which may be found ; that is, if any circle be defcribed through . 

 H and A, to meet the given lines in D, E, the difference between . 

 AD and AE is equal to N the given line. 



Join AH, BH, CH, DH. The triangles HCE, HBD are equal 

 to one another in every refpedl, for if BC be joined, the angle 

 HBC is equal to HAL, that is, by conftrudlion to HAB, there- 

 fore HB is equal to HC; in the fame way it appears that HD 

 is equal to HE ; now, the angle DHE is, equal to DAE, that is to 

 BHC, therefore BHD is equal to CHE, hence BD is equal to 

 CE, and the difference between DA and AE is the fame with . 

 the difference between BA, AC, which by conftruiftion is equal , 

 to the given line M. 



These two laft propolitions may be confidered as particular 

 cafes of the following propoiition, 



PROP, IV. \ 



