86 The Three Sections, Tangencies, 



point Q in it can he found, such that I being any point in IjL, 

 and E, F the points in which IB, IC cut MM 07id NNj then 

 wilt PE.QF be of a given magnitude m.n. 



For PBG^ BU^ and BH are known, as also HC; and CD 

 parallel LL is known in position; and .'. since PU.QD = 

 m.n, it is e^ddent NDQN between HC and CD and parallel 

 to CG is known in position. 



If m.n is not restricted in sign, there are two answerable 

 lines NN equidistant from CG, &c. 



POBISM. 



Given the points V, Q, in the given lines MM, NN, and 

 given also a point B and a straight line LL in position; 

 another point C can be found, such that I being any point in 

 LL, and E and F the points in which IB and IC cut MM and 

 NN, we shall have PE.QF of a constant determinable magni- 

 tude. 



For PBG, BH, and HQ are known, and therefore the point 

 C where GC parallel to NN cuts HQ, and therefore also 

 PE.QF = PU.QD is determined. 



PORISM. 



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

 and NN, and given also a point B ; a point C and a straight 

 line JAa, parallel to a given direction can be found, such that I 

 being any point whatever in this line, and E, F the points in 

 which IB and IC cut MM and NN, we shall have PE.QF of a 

 given magnitude m.n. 



The straight line PBG is known in position, and the line 

 BH parallel to MM is known in position ; and QA and point 

 A are known ; and drawing BU parallel to the given direc- 

 tion, the point U in MM is known, and therefore D in NN 

 (because PU.QD = PB.QA = m.n) ; therefore DC parallel 

 to BU is known in position. And since if through A we 

 draw AD' parallel to CD (since angle AC right to D' = CA 

 right to D = BC right to G) the circle ACB is known ; hence 

 the point C in which this circle cuts DC is known, and 

 therefore CG parallel to NN, and the point G in which it 

 cuts PB, and therefore LL through G parallel to DC or BU. 



It is evident there are two answerable points C, and that to 

 each one corresponds an answerable line LL. 



