and Elementary Theorems of Geometry. 93 



any two lines making the angle OB right to C = 9 right, and 

 cutting MM and NN in E and F, we shall have PE ■ QE = 

 a constant determinable magnitude. 



For the circle BCO is determinable, as also PB, QC, and 

 points L and H, and therefore MM and NN, through P and 

 Q, parallels to BH and CL. And we have the rectangle 

 PE • QF equal PB ■ QC. 



PORISM. 



Given the points P and Q in the given straight lines MM 

 and NN, and given any other point B in the same plane ; a 

 point C may be found, and an angular magnitude 9 right, 

 such that BO and CO being any two lines making angle OB 

 right to C = 9 right, the points E and F, in ivhich they cut 

 MM and NN, arc so related that PE ■ QF shall be of any 

 given magnitude m ■ n. 



For the line PB<7 is determinable, and .". circle P<?Q, and 

 hence point p ; and, since PB ■ QC must be equal m . n, the 

 point C is determinable. The angle 6 right is evidently equal 

 PM right to B. 



PORISM. 



Given an angle 9 right, and the points P and Q in given 

 lines MM NN; two points BC may be found on any curve 

 or line, such that BO and CO making the angle OB right to 

 C = 9 right, and cutting the given lines in E and P, toe shall 

 PE . QF of a constant determinable magnitude. 



For the angle 9 right being given, Vq and Qip are deter- 

 minable, and the points B and C in which they cut the given 

 curve or line, "and therefore PB . QC, or the magnitude of PE 

 BF. 



The points BC or one of them may be sometimes imaginary, 

 &c. 



PORISM. 



Given the straight lines MM NN, and the point P and Q 

 therein ; two points B and C may be found in the circumfer- 

 ence of a given circle, such that O being any other point 

 whatever in the circumference, the lines OB and OC shall cut 

 MM and LL in points E and F so that PE . QF will be of 

 a constant determinable magnitude. 



Here we have to draw PBL and QCH so that PH 

 and QL may be parallels to MM and NN respectively. 

 The points B ■ and C thus determined give PE . QF = 

 PB . QC— The method of drawing PBL and QCH as indi- 

 cated is obvious. For drawing LS parallel to PH or MM to 



