and Elementary Theorems of Geometry. 81 



tances from the starting point to any of the others, and from 

 this last to another, and so on consecutively, as may be, 

 until a final arrival at the starting point be equal zero. Cor. 



If a b c he any three points in a straight line, then will 

 ab + be + ca = zero. 



And from this we can generalise Euclid's Second Book. 



Given any number of straight lines AA, BB, CC, DD, 

 EE, FF, &c, if one of each of the following pairs of angles 

 be given, viz. : — Either AA right or left to BB, BB right or 

 left to CC, CC right or left to DD, DD right or left to EE, 

 EE .right or left to FF, &c, then BB and EE being any two of 

 these lines, the angles BB right and left to DD will be given. 



If BB and AA be parallel straight lines, and CC any 

 straight line intersecting them, then will the angle CC right 

 to AA = angle CC right to BB, and angle CC left to AA = 

 CC left to BB, and angle AA right to CC - BB right to CC, 

 and AA left to CC = BB left to CC. And reciprocally, if 

 any one of these relations exists amongst the angles made by 

 CC with BB and AA, then will BB and AA be parallels. 



If there be any two intersecting straight lines, and that on 

 each of them there is a pair of points, such that the rectangle 

 under the distances of the point of intersection from the 

 points on the first line shall have the same magnitude and 

 sign as the rectangle under the distances of the intersection 

 from the points on the second line, then will these two pair 

 of points lie in the circumference of a circle. And recipro- 

 cally, &c. 



If ABCD be any four points so related that the angle 

 AC right to B = angle DC right to B, or that AC left to 

 B = DC left to B, then will these four points lie in the cir- 

 cumference of a circle. And reciprocally, &c. 



If ABCD be four points such that angle AB right or 

 left to D is equal angle CB right or left to A, then will the 

 line AD touch the circle ACB in A. And reciprocally, &c. 



If BA be fixed points, and C a point so restricted that 

 the angle CA right or left to B is of a constant magnitude, 

 then will the locus of C be the circumference of a fixed circle 

 through the points A and B. And the reciprocal is also true. 



If aa and bb are fixed straight lines through fixed points 

 -A and B, should cc and dd be any other pair of straight 

 lines through A and B, making the angle cc right or left to 

 aa equal to the angle dd right or left to bb, then will the 

 locus of the intersection of cc and dd be a fixed circle passing 

 through A and B. 



