The Imaginary m Geometry 



25 



Here the expression in the first parenthesis is the distance from 

 F 3 to the line PJP 2 multiplied by V (1 +*» 2 )> while x\ — x 2 is the 

 distance P 1 P 2 divided by VC 1 +w 2 ). Evidently the triangle is 

 finite and not zero if the vertices all lie in the finite region and are 

 non-collinear. In particular, suppose the points are (o, o), (1, i), 

 (2, i), of which the first two are on a circular ray at no distance 

 from each other, but at an infinite distance from the third ; the 

 area is — i/2. This may be verified by noticing that the last two 

 are on y = i, which is distant i from (o, o). 



Fig. 25 shows how the area is made up in the general case. 

 The real part is the triangle P/PV-^Y -\- 3 quadrilaterals of the 



Fig. 25. 



type P Z 'P 2 P & P 2 . The imaginary part is the sum of three quadri- 

 laterals of the type P 2 T^P^P X . The area of P Z 'P 2 P 3 P 2 ' is the 

 opposite of that of P Z 'P 2 "P S "P 2 ', which is 



Thus the sum of the three areas of the type is 



*i" .V 1 



x.r y," 1 



A- " A. " T 



25 



