100 GEOMETRICAL RESEARCHES, 



taken in respect to any point in the circumference. And 

 from these we have 



b 



n 



the rotatives of the involved parts being taken in respect to any 

 point in the circumference. 



Now since the product of two sides of any plane triangle is 

 equal in magnitude to the product of the diameter of the circum- 

 scribing circle and the perpendicular from their point of inter- 

 section on the third side ; it is evident from the last equation that 

 the perpendiculars from p l on the straight lines a l b l and b l 

 a have to each other a known numerical ratio ; and, therefore, 

 all the answerable positions for p l must be included amongst 

 those given by the intersections of the given circle with two 

 known straight lines passing through the intersection of the 

 chords a b and b a . Moreover, since one only of 



1 W-f 1 * M pi 



these two straight lines gives points jp l fulfilling the equation 

 of conditions when the rotatives of the involved portions 

 are taken in respect to any point in the circumference, it is 

 obvious that the point p l is an intersection of the given circle 

 with one known straight line. 



By forming another inscribed w'gon c l c 2 .... c , x whose sides 

 pass in order through the given points, it is evident the point p l 

 must be in a deter minable straight line through the intersection 

 of the chords a l c n+1 and c l a . And hence we infer Poncelet's 

 method of finding the straight line x x containing the answerable 

 positions 



When a n+l and b are co-incident with a l and b l then 

 evidently the first side of the last equation is equal unity, and 

 therefore so also the second side. And in this state of the data 

 it is evident that any point in the circumference will be an 



