GEOMETRICAL F R I S M S. 119 



in V, which will be a given pointy fmce GA, AD, are given by 

 pofition. 



Join PV, the angle PVA is equal to PAE or PDS, that is, 

 (P, D, M, N being in a circle) to PNM, and PDV is equal to 

 PMN, the triangle PMN is therefore fnnilar to PDV ; and fmce 

 the angle PVA is equal to PDS, alfo PNV to FMD, the triangles 

 PDM, PVN are fimilar. Thus it appears, that HN and AV 

 are fimilarly divided by the lines BK, Cli, DM, &c. ; now, the 

 points A, B, C, D, V, l^c. are given ; therefore the ratios of 

 HK, KL, LM, MN, ^r. to one another are given. CX^ E. D. 



Cor. I. The lines PH, PK, PL, PM, PN, l^c. contain given 

 angles, and have to each other the given ratios of PA, PB, PC, 

 PD, PV, ^r. 



CoR. 2. The line HN cuts off from the given lines, fegments 

 HA, KB, LC, DM, VN, 'zJc. adjacent to given points, and hav- 

 ing alfo to one another the given ratios of PA, PB, PC, PD, 

 PV, ^c. ; for the triangles PAH, PBK, PCL, PDM, PVN, ^c 

 have been proved equiangular ; and therefore AH, BK, CL, 

 DM, VN, 6'c. are proportional to PA, PB, PC, PD, PV, ^c. * 



PROP. VIIL 



* It may be proper to remark here, that, in the preceding propofitions, the 

 ftraight lines given by pofition, as well as the indeterminate flraight line, which is 

 cut by them into fegments, having to each other given ratios, and which alfo cuts 

 off from them fegments adjacent to given points, and having to each other given ra- 

 tios, are tangents to a parabola, of which the point that is required to be found is the 

 focus. This confideration fuggefts fome curious propofitions, relating to tangents to. 

 the parabola. Some of them have been obferved by Dr Halley, in his tranflation 

 of the SeBio Rationis of Appollonius. 



One very obvious application of the propofitions above hinted at, is to defcribe 

 parabolas that fliall pafs through given points, and touch ftraight lines given by poii- 

 tion. 



