112 GEOMETRICAL PORISMS. 



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



-Two flraigKt 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 defcribed through it 

 and A the interfe6lion of the given lines, to meet them in 

 D, E, Ihall cut off from them fegments AD, AE, fo that 

 PXAD + QXAE, fliall be equal to a given fpace. Alfo, the 

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

 fo that PX AD — Q2< AE, fliall be equal to a given fpace. 



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

 the hypothefis of the propofition, or fo that PX AB-f-Q^X AC, 

 ( fig. 5. ) may be equal to PXAD+Q_XAE, and fo hat 

 PXAB— Q^XAC maybe equal to PXAD— QjKAE, (fig. 4.) 

 then, in both cafes, P X BD will be equal to Q^X CE ; 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 

 diredled, and if PX AD-f-QX AE is to be a given fpace, (fig. 5.) 

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

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

 may lie in contrary directions 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 PX AB-|-Qj><AE to 

 each, we get PXAD-f QxAE, equal to PXAB + QX AC, that 

 is, to the given fpace, as was required. 



But if PX AD — Qj<AE is to be a given fpace, (fig. 4.) find 

 H, (Prop. I.) fo that any circle pafiing 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 



QXCE, 



