40 ANM.YTH- OEOMEl i, ll. 



The same result \\..ul.l have been obtain. -<l lial HI, - :i. instead of 

 m t = 4, been given the minus sign ; or, again, formulas [7] could have 

 been employed to solve this problem. 



5. Solve Ex. 4 directly from a li^nn-. without using either [6] or [7]. 



6. Find tln |oints which divide the line frotn(l. inter- 

 ually and externally into segments which arc in the ratio 2 



7. A line AB is produced to C, so that BC = } AB\ if the points A 

 n have the coordinates (5, 6) and (7, 2), respectively, what are the 



coordinates of C? 



8. Prove. l>y means of Art. 30, that the in. -(linn lines of a triangle 

 meet in a |><>ini. \\ hi.-h is for each median the point of trisection nearest 

 the side of the triangle. 



31. Fundamental problems of analytic geometry. The. 

 elementary applications already considered have indicated 

 how algebra may be applied to the solution of geometric 

 problems. Points in a plane have been identified with pairs 

 of numbers, the coordinates of those points, ;m<l ii 

 been seen that definite relations between such points corre- 

 spond to definite relations between their coordinates. 



It will be found also that the relation between points. 

 which consists in their lying on a definite curve, com - 

 s| Minds to the relation between their coiinlinatcs, which 

 consists in their satisfying a definite equation. From this 

 arise the two fundamental problems of analytic geom- 

 etry : 



I. Given an equation, to find the corresponding geometric 

 curve, or locus. 



II. Given a geometric curve, to find the correspond i/<;/ 



When tliis relation between a curve and its equation lias 

 been stmlinl. then a third problem arises : 



III. To find the properties of the curve from those of its 



