780 PROCEEDINGS OF THE AMERICAN ACADEMY. 



Thus distance is the moment of Euclidean distance with respect to F 

 and angle is the reciprocal of a moment. 



It is to be observed that the sum of the three sides of a triangle is 

 twice the Euclidean area. In fact, 



Figure 27. 

 AB + BC + CA = 2 (ABF + BCF + CAF) = 2 ABC. 



This is another justification for calling the sum of the sides of a triangle 

 the area of the triangle. 



Case II. The Point F on the Line/. 



37. In this case distances are equal if they project from F into equal 

 vectors on a line. All triangles having vertices A, B, C respectively 

 on fixed lines passing through F have equal sides, but may have differ- 

 ent angles. Hence in this case it is not possible to express an angle 

 in terms of the sides as was done in Case I. 



Furthermore (Figure 28). 



AB + BC = AD + DC = AC. 



Similar relations hold for angles. We thus have two equations con- 

 necting the parts of a triangle. 



« + ^ + c = 0, 



(24) 



We should expect one other relation. This may be found as follows: 

 Let AB, BC, and CA cut f in P, Q, R. Then 



| = g = -(DA|CIl)=-(FP|QR). 



