36 The Three Sections, Tangencies, 



GENERATING PROBLEM TO THE THREE SECTIONS. 



Given the points V Q, on a straight line MM^ and the points 

 'R ^ on a straight line NN ; through two given jyoints A B 

 to draw two straight lines Ki, V>1, making the angle I A right 

 to'B of a given angular magnitude right, and such that C and 

 D being the points in ivhich AI and BI cut JMM and NN, ^ve 

 shall have PCS!) : QC.RD : : I : k; ( j, behig of given mag- 

 jiitude and known sign in respect to the directions on MM and 

 NN). 



ANALYSIS. 



The circle AIB is evidently kuown. Suppose we draAV a 

 straight line CEF, making the angle CE or CF right to P or 

 Q equal the angle DB right to U or S^ and tliat through P 

 and Q, we draw PE and QF to cut it^ so that tlie angle PE 

 riglit to C = EB riglit to D, and tlie angle QE right to C 

 = SB right to D. Then the triangles CPE, CQE are similar 

 to the triangles DRB, DSB ; moreover it is evident that from 

 these triangles we have the relation PE.SB : QF.EB : : 

 PC.SD : QC.PvD :: I : k. 



Let G be the point of intersection of PE and QF. 



From the last proportion we have PE to QF in the known 

 ratio of /.BB to /c.SB ; and therefore (see Porism 4th, in 

 Transactions of 1859) the circles EGF, CQF, CPE pass 

 through a point O in the circumference of the circle PGQ, 

 which is such that PO : QO : : /.EB : /c.SB; and hence, as 

 PG and QG are known, the point O in the circle PQG is 

 known. 



Again, the angle OC right to P = EC right to P = BD 



