122 GEOMETRICAL PORISMS, 



Pc, PD, Vd, l^c. Becaufe the points P, E, A, B, are in a circle, 

 the angle PA« is equal to PBZ' ; now VaK is equal to PZ'B ; for 

 P/aE is equal to P3E, the triangles P<7A, P3B are therefore limi- 

 lar. In the fame manner it may be fhewn, that VUE is iimilar 

 to P<rC, and that again to VdD, &c. Therefore PA is to PB as 

 Ta to P^, and PB to PC as P^ to Pc, and PC to PD as Vc to 

 Td,.l^c\ ; now the angles APB, BPC, CPD, i^c. are equal to 

 AEB, BHC, CKD, £5fc. that is, to a?b, 6Fc, cVd, l^c. therefore 

 if ab, hcy cdf l^c. Ad be joined, the re<5lilineal figure PABCD, 'is'c. 

 is fimilar to Pa bed, l^c ; and leaving out the fimilar triangles 

 PAD, Pad, the redlilineal figure ABCD, ^c. is fimilar to abed, 

 <3'c. Now the points P, E, /?, being given, the circle pafling 

 through them is given ; therefore b is b. given point ; in Hke 

 manners, d, i^c. are given points; therefore the figure abcdj l^c. 

 is given ; therefore ABCD, l^c. to which it is fimilar, is given 

 in fpecies. Q^ E. D. 



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

 gles, and have to each other the given ratios of P^, P^, Pc, Pd, 



CoR. 2. The fegments Aa, Bb, Cc, 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 Pa, Pb, Pf , Pd, tsfc, 



CoR. 3. If there be any number of flraight lines given by 

 pofition, there may be innumerable redlilineal figures fimilar 

 to one another, and having their angles upon the flraight lines 

 given by pofition. 



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



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

 given circle. Let C be a given point in KC, a flraight line 



given 



