GEOMETRICAL P ORIS MS. 129 



which render the problem indeterminate, are thofe which are 

 required to be found, in tlie proposition jufl now quoted. 



PROP. XIV. PROBLEM, Fig. 19. PI. IV. 



Three ftraight lines AB, AC, BD are given by pofition, and 

 P is a given point. It is reqiiired to draw PE to meet BD 

 in E, and PG meeting AB in F, and AC in G, fo that the 

 angle EPG may be given, and fo that EP may have to 

 jFG the giv«n ratio of a, to (B. 



Suppose the lines drawn as required. In GP take PH equal 

 to FG, therefore the ratio of EP to PH will be given, now the 

 angle EPH is given, tlaerefore H is in a ftraight line given by 

 pofition, (ApoU. Loci Plani, Lib. i. Prop. 6.) let this line be 

 LC. Bifetft PF in K, then becaufe P is a given point, and 

 AB is given by pofition, the point K will be in a ftraight fine 

 given by pofition, (Loci Plani, Lib. i. Prop. 4.) let this line be 

 LM, Becaufe GF is equal to PH, and FK. to PK, therefore 

 GK is equal to KH, but the lines ML, MC, CL are given by 

 pofition, therefore, (Prop. 5.) a given point N may be found in 

 the circumference of a circle pafTmg through M, C, L, fuch, that 

 the points N, M, G, K are in a circle, therefore if this point be 

 found, and NG, NM joined, the angle NGK or NGP is equal 

 to the given angle NML, now N and P are given points, there- 

 fore G is in the circumference of a given circle, but it is alfo 

 in a flxaight line given by pofition, therefore the point G is 

 given. 



Construction. Find LC a ftraight line given by pofition, 

 fuch, that if PE, PH be drawn meeting BD, CL, and contain- 

 ing an angle EPH equal to the fupplement of the given angle 



