222 KANSAS ACADEMY OF SCIENCE. 



Now, two triangles, ABC and A'B'C, in which AA', BB', CC 

 produced are concurrent may always be considered as the projection 

 of a triangular pyramid cut by two oblique planes in ABC and A'B'C. 

 Thus the foregoing proposition is established. In a similar manner 

 the converse proposition may be proved. 



As a second example, let us take the proposition concerning three 

 circles in a plane : 



The external centers of similitude of three circles of a pl&ne are 

 collinear. Any two internal centers of similitude are always collinear 

 with one of the external centers. 



Fig. 2. 



To prove this, let us consider three spheres whose projections are 

 the three given circles in the plane. Any two of these spheres admit 

 of an internal and external common tangent cone. Thus, designating 

 the centers of the spheres by Ci, C2, C3, we obtain three external and 

 three internal tangent cones whose respective vertices E]2, Eaa, E31, 

 and I12, I23, I31, are coplanar, the plane P passing through Ci, C2, C3. 

 The three spheres and the six cones have the same common tangent 

 planes and these are 2 by 2 symmetrical with respect to P. Any 

 two symmetrical planes ( there are eight common tangent planes ) are 

 common tangent planes to the three spheres- and to three of the 

 tangent cones. 



The three vertices of these cones are therefore necessarily collinear. 

 We have therefore the result that the six vertices of the common 

 tangent cones are coplanar and are 3 by 3 situated in straight lines; 

 /. e., they form a complete quadrilateral. 



The collinear groups are, fig. 2 : 



E12 E12 E23 E31 



B23 I23 I31 1 12 



E31 1 31 1 12 I23 



