Plane Geometry - Bruce E. Shapiro

SECTION 14. BETWEENNESS 65which verifies property (1).To get property (2), observe that the argument of the square root is anon-negative number, hence the square root is defined and positive or zero.For property (3), first assume P = Q. Thend(P, Q) = d(P, P )»= (x 1 − x 1 ) 2 − (y 1 − y 1 ) 2= 0This shows that (P = Q) ⇒ (d(P, Q) = 0), To get the converse, assumethat d(P, Q) = 0. Then»0 = (x 1 − x 2 ) 2 + (y 1 − y 2 ) 20 = (x 1 − x 2 ) 2 + (y 1 − y 2 ) 2 (14.1)If either x 1 − x 2 ≠ 0 or y 1 − y 2 ≠ 0 then the right hand side of equation14.1 is non-zero. Hence x 1 = x 2 and y 1 = y 2 , which means P = Q. Thus(d(P, Q) = 0) ⇒ (P = Q).Example 14.2 (Cartesian Coordinates in the Euclidean Metric)In the Cartesian Plane any (non-vertical) line l can be described by someequationy = mx + band any vertical line byLetif l is non-vertical, and setif l is vertical.y = af(x, y) = x √ 1 + m 2 (14.2)f(x, y) = f(a, y) = yTo see that f is a coordinate function and that this works, we need toconsider each case (vertical and non-vertical) separately, and to show thatf is 1-1, onto, and satisfiesin each case.P Q = |f(P ) − f(Q)| (14.3)Suppose first that l is non-vertical, and define f as given by equation 14.2.Revised: 18 Nov 2012 « CC BY-NC-ND 3.0.