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UNIVERSITY OF COLORADO STUDIES 



3. If the sides of a qitadrilateral u^u^u^u„ inscribed to a cir- 

 cle^ alternately pass through two fixed points -y,, v^of a straight line 

 Z, Fig. 3, then there are an infinite nutnber of inscribed quadri- 

 laterals u^' u^ u^' u^ ^ u^' 'u^ 'u^ 'u^ ' ^ . . . with the same property. 

 The corresponding diagonals of all these quadrilaterals pass 

 through a fixed pointy the pole of v{v^. 



Pig. 3. 



These theorems, to which the reader may add a great number of 

 others, may, of course, immediately be duplicated by the principle 

 of duality. It is well to remark that the first and third of these 

 theorems are also special cases of the problem, to inscribe a polygon 

 to a conic whose sides, in a certain order, pass through fixed points 

 of the plane, which in case of a triangle and circle was already 

 solved by Castillion and Lagrange in 1776. The extension of the 

 problem to polygons was given by Oltajano and Malfatti, For 

 the triangle the extension to conies was given by Brianchon and 



