112 GEOMETRICAL PO'RISMS. 



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



Two ftraiglit lines AB, AC being given by pofition, and two 

 lines P, Q^ being given in magnitude, a point H may be 

 found, (fig. 5.) fuch, that any circle defcrlbed through it 

 and A the Interfecflion of the given lines, to meet them in 

 D, E, ihall ciit off from them fegments AD, AE, fo that 

 PXAD + Q2<AE, fliall be equal to a given fpace. Alfo, the 

 fame things being fuppofed, a point H may be found, (fig. 4.) 

 fo that PXAD— QxAE, Ihall be equal to a given fpace. 



Let given points B, C, be taken in either cafe agreeing with 

 the hypothefis of the propofitlon, or fo that PxAB + Q^xAC, 

 (fig. 5. ) may be equal to PXAD+Q^XAE, and fo hat 

 PXAB— C)_XAC maybe equal to PXAD— Qj<AE, (fig. 4.) 

 then, in both cafes, PxBD will be equal to Q^XCE ; there- 

 fore BD is to CE as Q_to P, that Is, In a given ratio, and the 

 points B, C being given, the point H may be found, (Prop. i.). 



Construction. Let given points B, C be taken as above 

 dlreded, and if PX AD+QX AE is to be a given fpace, (fig. 5.) 

 find a point H, (Prop, i.) fo that any circle defcrlbed through 

 A and H may meet the given lines in D, E, fo that BD, CE 

 may He in contrary dlredtions to AB, AC, and have to each 

 other the given ratio of Q^ to P, then P X BD will be equal to 

 QxCE, and adding the common fpace PXAB-fQ_XAE to 

 each, we get PX AD-}-Q2< AE, equal to PXAB + QX AC, that 

 Is, to the given fpace, as was requli"ed. 



But If PxAD — Q^AE is to be a given fpace, (fig. 4.) find 

 H, (Prop. I.) fo that any circle paffing through H, A may cut 

 off fegments BD, CE, in the given ratio of Q_to P, and lying 

 ■towards the fame parts with AB, AC, then PXBD is equal to 



Q2<CE, 



I 



