Lazy tongs and Lattice-works. 379 



If now two straight rods or lines, AB, CD (fig. 2) are taken 

 in one plane, and meeting in E (whether in their actual lengths 

 as shown, or in their geometrical productions, is immaterial), 

 and so conditioned that A, C, B, D lie in one circle, then the 

 products o£ their segments are equal or CE . ED = AE . EB. 



Ffr. 2. 



It follows that E remaining the same for both rods 

 A, C, B, D will always lie on a circle, whatever be the 

 angle between the rods. 



If DE=n.AE, and CE = ?n.AE. then the condition is 

 fulfilled if EB = m?i.AE ; n and m may have any values 

 whatever. 



It will be convenient for geometrical reasoning to imagine 

 or describe the straight lines AC, CB, BD, DA. Then the 

 triangles DEB, AEC, are similar, and 



DE : EB : BD :: AE : EC : CA :: n : 1. 



Similarly regarding the triangles CEB, AED, they are 

 similar, and CE : EB : BC :: AE : ED ; DA :: m : 1. 



Call the angles ECA, EAC, EAD, EDA, a, £, 7, and 8 

 respectively. Then also the angles EBD, EDB, ECB, EBC 

 are », /3, 7, and 8 respectively. 



Now DB may be derived from AC, as regards direction 

 and magnitude, by allowing AC to revolve first through ACE 

 or a in one direction, and then through EDB, or ft in the 

 opposite direction, and by reduction in the ratio 1 : n. 



Thus DB makes with AC the angle a —0, and DB=?i . AC. 



Similarly CB makes with AD the angle 7 — 8 and 

 CB = m.AD. 



Now suppose another pair of rods DF, BGr jointed at H, 

 and similar to the first pair in all respects but bearing the 

 ratio to them of n : 1, jointed on to the first pair at B and D. 



