Existence Postulate: The collection of all points forms a nonempty set. There is more than one point in that set.

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1 Undefined Terms: Point Line Distance Half-plane Angle measure Area (later) Axioms: Existence Postulate: The collection of all points forms a nonempty set. There is more than one point in that set. Incidence Postulate: Every line is a set of points. For every pair of distinct points A and B there is exactly one line l such that A l and B l. Notation: Since the Incidence Postulate makes it cle ar that there is a unique line through any two points A and B, we will denote that line by AB. We often say that A and B determine the line AB. Definition: The statements P lies on l, P is incident with l, and l is incident with P all mean that P l. For the purposes of this class we will also say P is on l and l contains P and probably other things that communicate the same basic meaning. Definition: P is said to be an external point for line l if P does not lie on l. Definition: Two lines l and m are said to be parallel, written l m, if there is no point P such that P lies on both l and m. In other words, the lines l and m are disjoint as sets of points. Theorem: If l and m are two distinct nonparallel lines, then there is exactly one point P such that P lies on both l and m. This is a basic theorem of Incidence Geometry, proved as Theorem in Chapter 3. Note: This theorem implies that for any two lines l and m, either l = m, l m,or l m consists of exactly one point.

2 The Ruler Postulate: For every pair of points P and Q there exists a real number PQ, called the distance from P to Q. For each line l there is a one-to-one correspondence from l to such that if P and Q are points on the line l that correspond to real numbers x and y respectively, then PQ = x y. Definition: Given three distinct points A, B, and C. The point C is said to be between A and B, written A C B, provided C ABand AC + CB = AB. Note that there are two parts to betweeness: one about collinearity, the other about distance. Definition: The segment AB = { A, B} { P A P B}. The ray AB = AB { P A B P}. Definition: The points A and B are the endpoints of the segment AB, and the other points are interior points. The point A is the endpoint of the ray AB, and the other points on the ray are interior points. Definition: The length of the segment AB is AB, the distance from A to B. Definition: Two segments provided AB = CD. AB & CDare said to be congruent, written AB CD, Theorem: If P and Q are any points, then: 1. PQ = QP. PQ 0, and 3. PQ = 0 if and only if P = Q. (Pre-proof planning: The best tool OK, at this point the ONLY tool we have for dealing with distance is the ruler postulate. In order to use it we need to get the points on a line. So our plan is first to get P and Q together on a line, then use the ruler postulate. Since part 3 of the theorem deals with the possibility that P and Q are the same, we also need to take care of that in our proof whenever the possibility arises.)

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4 Corollary: A B C if and only if C B A Notation: Let the set of all points be denoted by. Definition: A metric 1 is a function D : [0, ) such that for all points P and Q: 1. DQP (, ) = DPQ (, ). DPQ (, ) 0 3. DPQ (, ) = 0 if and only if P= Q Definition: Let l be a line. A one-to-one correspondence f : l such that PQ = f ( P) f ( Q) for every P and Q on l is called a coordinate function for the line l. The number f ( P ) is called the coordinate of the point P. Note that with these two definitions in place, the Ruler Postulate simply states that there is a metric in our geometry, and that every line has a coordinate function. We look at some examples of metrics and coordinate functions next. 1 The definition of a metric generally includes the triangle inequality, DPQ (, ) DPR (, ) + DRQ (, ) for all points P, Q and R. We don t include it because we can prove it follows from future axioms.

5 Examples: We look at two models in which points are sets of ordered pairs of real numbers, and lines are sets of points satisfying linear equations: Points: {( x,y) x, y } Lines: {( x, y) x, y, ax + by + c = 0; a, b, c ; a 0 or b 0} We define two different metrics: Euclidian Metric: For points P= ( x1, y1) and Q = ( x, y), PQ = ( x x ) + ( y y ) 1 1 Taxicab Metric: For points P= ( x1, y1) and Q = ( x, y), PQ* = x x + y y P (x 1, y 1 ) 6 y - y 1 4 Q (x, y ) x - x 1 5 -

6 Coordinate Function in the Euclidean metric: Given a line y = mx+b, we can write two points P(x 1, y 1 ) and Q(x, y ) on the line as (x 1, mx 1 + b) and (x, mx + b). Then PQ = ( x x ) + (( mx + b) ( mx + b) ) 1 1 PQ = ( x x ) + m ( x x ) 1 1 PQ = (1 + m )( x x ) 1 1 PQ = (1 + m ) x x PQ = k x x 1 Since PQ is supposed to be the absolute value of the difference of ruler coordinates, we can take these ruler coordinates to be the x-coordinates of the point, multiplied by a constant that depends only on the slope of the line. So, given the Euclidean Metric, we can define a coordinate function f for the line y = mx+ b as follows: For all points (, ) x y lying on the line, define (, ) ( (1 )) f xy= + m x. It is very easy to show that f is one-to-one and onto. Moreover, if P(x 1, y 1 ) and Q(x, y ) are two points on the line, we just showed that f ( P) f( Q) = (1 + m ) x (1 + m ) x = (1 + ) 1 = m x x PQ 1 So this is a coordinate function for the Cartesian Plane with the Euclidean metric.

7 Coordinate function in the Taxicab metric: If we again consider two points P(x 1, y 1 ) and Q(x, y ) on a line y = mx+ b, we get: PQ * = x x + ( mx + b) ( mx + b) 1 1 PQ * = x x + m x x 1 1 PQ * = (1 + m ) x x PQ * = k ' x x 1 1 In this case, we can define a coordinate function as follows: For all points ( x, y) f ( xy, ) = 1 + m x. on the line, define ( ) lying In exactly the same way, we can show this is a coordinate function for any line with slope m in the Cartesian Plane with the Taxicab metric. In other words, the ruler postulate works here, too, with just a different choice of constant. Note that since k PQ* = PQ, k distance in one model can be determined from distance in the other simply by multiplying by constants that depend only on the slope of the line containing the two points. Moreover, betweenness is exactly the same for the two models, since ruler coordinates and distance in one are just a constant (depending on slope) times ruler coordinates and distance in the other. Note: In the above calculations we have ignored the case where slope is undefined, that is, vertical lines. You can fill in those straightforward details.

8 Theorem (The Ruler Placement Postulate): For every pair of distinct points P and Q, there is a coordinate function f : PQ such that f( P ) = 0 and f ( Q ) > 0. This theorem allows us to establish a convenient coordinate function whenever we wish. Thus we can effectively make any line into a number line where we get to choose the zero point and which direction is positive. As the name implies, this is often adopted as an axiom or postulate (or as part of the Ruler Postulate) just to make life easier (e.g. avoiding the work of this theorem and of Exercise 5.5). Finally note that we have now shown that there are two models of the axiom system we have built up to this point: The Cartesian Plane with the Euclidean Metric The Cartesian Plane with the Taxicab Metric. Among other things, this shows that the axiom system is consistent (at least as consistent as our usual Cartesian plane and distance). Later, we will use the fact that both of these are models to show than a new axiom is independent.