776 PROCEEDINGS OF THE AMERICAN ACADEMY. 



If from a point in the plane of a triangle lines a', b', c' are drawn 

 to the vertices A, B, C respectively, then 



c'a a'b b'c _ 

 c'b a'c b'a 



and conversely, if the above relation holds, the three lines a', b', c' pass 

 through the same point. 



If any line b iyi the plane of the triangle abc cuts the three sides in 

 A', B\ C\ respectively, and the lines a', b', c' are drawn from these 

 points to the opposite vertices, then 



a'c c'b b'a _ 



a'b c'a b'c 



and conversely, if this relation holds, the three points A', B\ C' lie on 

 a line. 



From these theorems result many propositions analogous to those of 

 ordinary geometry: 



Lines intersecting the base of a triangle on f divide the other two sides 

 into jjropmrt tonal parts. (It should not be expected that the bases of 

 the triangles thus formed have the same ratio, for this would be equiva- 

 lent to symmetry which does not exist in this geometry.) 



If one of the vertices of a triangle is joined to F and lines are drawn 

 from any point mi this line to the other vertices these li?ies divide the 

 angles into proportional jyarts. 



Lines drawn from the vertices to the mid points of the opposite sides 

 of a triangle meet in a point which is a point of trisection for the me- 

 dians, the longer part being toward the vertex. 



The bisectors of the angles of a triangle meet the opposite sides in three 

 p)oints on a straight line. It should be observed that the three bisec- 

 tors in this geometry cannot all cut the opposite sides between the 

 vertices. This is because the angle was defined to be that one which 

 does not contain F. 



If through any point in the plane of a triangle ABC, lines OA^ 

 OB, OC are drawn, intersecting the opposite sides in A', B', Cf ; and 

 if A'B', AB meet in G, B'C, BC in H and A'C, AC in K, tJie points 

 G, H, K lie on a straight line. 



35. Point-line Invariant. In this system we have nothing analo- 

 gous to perpendicularity, and there is no minimum distance from point 

 to line. If then we are to find a quantity analogous to the distance 

 from a point to a line in Euclidean geometry the analogy must have as 



