and Elementary Theorems of Geometry. 87 



the respective points in which these lines cut MM and NN, we 

 shall have PE to QF in a constant determinable ratio. 



The angle 9 right is evidently = angle MM right to NN, 

 &c. 



POKISM. 



Given two points P and Q, and given the two points BC 

 in the circumference of a given circle : then MM being an 

 arbitrarily assumed line through the point P, a Sne.NN pass- 

 ing through Q can be found, such that D being any point in 

 the given circumference, and EF those in which DB and DC 

 cut MM and NN, we shall have PE to QP in a constant 

 determinable ratio. 



The ratio is evidently that of PB to QC, &c. 



Given two points B C in a given circle, and a point P in a 

 given line MM ; another straight line NN and a point Q in it 

 can be found, such that D being any point in the given circle, 

 then will DB and DC cut MM and NN in points E and F, 

 making PE to QP in a given ratio m : n. 



For PBH and HCQ are in position, and the point Q 

 making QC : PB :: m : n, and QNN making angle QC right to 

 N = PB right to M. 



PORISM. 



Let P and Q be given points in the circumference of a circle; 

 if in lines from these points to any point H in circumference, 

 toe take PB and QC, having to each other a given ratio, then 

 ivill the circle BHC pass through a fixed determinable point O 

 in the given circumferance. 



It is evident O is such that PO : QO :: PB : QC, &c. 



And if the points PQ and circle be fixed only, then O is 

 fixed, &c. 



PORISM. 



Given two points PQ, in two lines MM NN, and given any 

 other point B ; another point C can be found, and a circle 

 through B and C described, such that D being any point in its 

 circumference, and EF those in which DB and DC cut MM 

 and NN, we shall have PE to QF in a given ratio of m to n. 

 . For PB is given, and therefore QC making angle QC right 

 to N = PB right to M is determinable in position; and 

 since QC : PB :: m : n, the point C is determinable. The point 

 H is the intersection of QC and PB, and therefore the circle 

 CHBD is determinable, &c- 



