and Loci of Apollonius, ^c. 85 



K'A must be parallel to NN, because the segment GI corre- 

 sponding is infinitely small. Therefore in the porismatic 

 state of the clata^ PBGK must be a straight line. And the 

 point Q must be such that after draAving DA making the 

 angle DA right to N = GL right to B, and DA.GB = DC. 

 GC, we shall have the angle QN right to A = PB right to 

 M, and QA.PB =: m.n. 



And for similar reasons to those which cause PB to pass 

 through G, will the straight line QC pass through the point 

 H in which BH parallel to MM cuts LL. 



Hence we easily deduce the following important porisms. 



PORISM. 



A point P fand but one) can be found in a given straight 

 line l^VSl, and a corresponding point Q. in another given line 

 NN, such that if straight lines BI, CI, be draivn from any 

 2ioint I in a third given line LL through tivo given points B, 

 C, to cut MINI and NN respectively in E and F, ive shall have 

 PE.QF of a constant determinable magnitude. 



For (from what has been observed) drawing BH and CG 

 parallels respectively to MM and NN, to cut LL in H and 

 G, it follows that GB will cut ]\IM in P, and HC will cut 

 NN in Q. And draAving CD and BU parallels to LL, to cut 

 NN and jNIM respectively in D and U, it is evident (since 

 these parallels intersect at infinity on LL) PE.QF must 

 always = PU.QD. 



PORISM. 



Given in position a straight line IAj and the points 1^ and C; 

 through two other given points P, Q, two straight lines MM 

 and NN (and but two) can be draivn, such that I being any 

 point in IAj and E and F the points in which IB and IC cut 

 MM and NN, loe shall have PE.QF of a constant determin- 

 able magnitude. 



For the lines PB and QC give us the points G and H in 

 their intersections with LL; and PMM, QNN, must be 

 parallels (respectively) to BH and GC. And PE.QF = 

 PU.QD = a known magnitude. 



PORISM. , 



Tkvo points B, C, and ttvcrstraight lines MM, LL, are given 

 in position, and a point P in MM ; a straight line NN and a 



