PHILLIPS AND MOORE. — ALGEBRA OF PROJECTIVE GEOMETRY. 



These relations solved for a, h, c give 



A+B+C 



a = 



b = 



c = 



BC ' 



A+B+C 

 AC ' 



A + B + C 

 AB ' 



769 



(14) 

 (15) 

 (16) 



FiGUKE 20. 



which have the same form as before as they should have by duality. 

 These equations can be solved for any three of the quantities A, B, C, 

 a, h, c in terms of the other three. Thus any three of these parts deter- 

 mine the triangle uniquely. In particular the three angles determine 

 the triangle and therefore similar triangles do not exist. 



We can construct a triangle having any three given lengths for sides. 

 In fact, take any segment BC of length a. Construct lines at distances 

 c, b from B and C respectively. These lines intersect in a point A such 

 that the triangle ABC has the sides required. There is no relation be- 

 tween the three sides of a triangle. Since any three parts determine 

 the triangle there is no relation between any three parts. Since we have 

 already found three independent equations connecting the six quantities 

 it follows that any other must be a consequence of these. 



VOL. XLVII. 



49 



