PHILLIPS AND MOORE. — ALGEBRA OF PROJECTIVE GEOMETRY. 771 



Consider the triangle AC'C. AC = — c', and angle AC'C = — C, where 

 C is the angle of the triangle AB'C. Then from relation (17) we have 



i- = p or b'C' = ha 







Then if we have the relation 



Area ABC a 



Area A'B'C ~ a" 



multiplying both numerator and denominator by the above equality 

 we have 



Area ABC ahC 



Area A'B'C db'C 



The two terms of the right member are equal to the sum of the sides 

 of the respective triangles. Hence 



Area A^B^C^ = «' _ <^' + ^' + c' 

 Area ABC ~ a ^ a -\- h -\- c ' 



Triangles having the same vertex and bases on the same line have line 

 areas proportional to their perimeters. 



Starting with the triangle ABC by a succession of operations consist- 

 ing of moving a side along its line and changing its length, we arrive 

 finally at any triangle A'B'C. Since under each of these operations the 

 area is changed in the same ratio as the sum of the sides, it follows that 

 the areas of any two triangles are proportional to their perimeters. We 

 choose the unit of area such that 



Area ABC = a + b + c. (18) 



That this expression for area has the properties required is evident 

 since it is determined by an ordered sequence of three points, and since 

 as already shown two triangles having the same vertex have perimeters 

 proportional to their bases. 



Perhaps the most fundamental property of area is the sum property, 

 i. e. that the area of a region is the sum of the areas of its parts. This 

 is a property of the sum a + b + c. For if we divide a triangle into 

 two triangles by a line through one of the vertices the sum of the pe- 

 rimeters of the two is equal to that of the original triangle since the divid- 

 ing line is counted twice in opposite directions. If then we define the 



