emch: involutoric collineations in plane and in space. 209 



symmetries whose axes pass t/iroi/g/i tlie point of intersection op tlic axis 

 of the one-termed group of linear elations and tlie axis of t lie symmetry. 



Fiti. 



In Fig. I these relations are all illustrated in the case of the 

 orthogonal symmetr}'', or the reflection on the X-axis. 



The original triangle ABC is reflected into A^BjC,. An elation 

 having its center at an infinite distance on the axis of Y, transforms 

 A^BjCj into a triangle AgBgCg, such that it forms an oblique 

 symmetr}- with the original triangle ABC. The axis S of the new 

 symmetry passes through the origin O. 



The purely geometrical proof is easily obtained from the figure. 

 It is well known that the areas of the three triangles ABC, 

 AjBjCj, AgBgCg are equal. 



{f) Central symjuetry is a special case of the perspective involu- 

 tion. It is represented by the equations 



The two-termed group of elations transforms the central symme- 

 try into the system of involutions 



a^x 

 ^ ^"bjX-f c^y— a/ 



Yi 



aiY 



b^x+Cjy— a/ 

 consisting of all involutions that leave the center invariant. 



