123 GEOMETRICAL P R I S M S. 



Vc, PD, P^, y^. Becaufe the points P, E, A, B, are in a circle, 

 the angle PA« is equal to PB/' ; now VaK is equal to PZiB ; for 

 PaE is equal to P3E, the triangles P/^A, P3B are therefore funi- 

 lar. In the fame manner it may be fhewn, that P3B is fimilar 

 to PrC, and that again to P^D, Wr. Therefore PA is to PB as 

 Va to P^, and PB to PC as Vb to ?c, and PC to PD as Vc to 

 Vd, i^c. ; now the angles APB, BPC, CPD, ^c. are equal to 

 AEB, BHC, CKD, "is^c. that is, to aVb, bVc, cVcl, isc. therefore 

 if ab, be, cd, l^c. Ad be joined, the redlilineal figure PxABCD, 'isfc. 

 is fimilar to Pabcd, ISc ; and leaving out the fimilar triangles 

 PAD, Yad, the redlilineal figure ABCD, ISc. is fimilar to abed, 

 y^. Now the points P, E, a, being given, the circle pafling 

 through them is given ; therefore Z" is a given point ; in like 

 manner i^, d, Isle, are given points; therefore the figure abcd,lSc. 

 is given ; therefore ABCD, Wr, to which it is fimilar, is given 

 in fpecies. Q^ E. D. 



CoR. I. The fines PA, PB, PC, PD, ^c, contain given an- 

 gles, and have to each other the given ratios of P^, Yb, Pr , Yd, 



CoR. 2. The fegments Ka, B3, Cr, D^, l^c. of the given 

 lines, adjacent to the given points a, b, c, d, l^c. have alfo to 

 each other the given ratios of Va, Vb, Pr , P^, ^c. 



CoR. 3. If there be any number of fl;raight lines given by 

 pofition, there may be innumerable redlilineal figures fimilar 

 to one another, and having their angles upon the fl;raight lines 

 given by pofition. 



PROP. X. PORISM, Fig. 15. PL III. 



Let a and B be two given points in the circumference of a 

 given circle. Let C be a ^iven point in KC, a llraight line 



given 



