Question 3 Section 3.2Distance and the Ruler Postulate39Corollary 3.2.8. A * C * Bif and only if B * C * A.Proof. Let A, B, and C be three points such that C E AB (hypothesis). If A * C * B,then AC + CB = AB (definition). Since ABthe case that BC + CA = BA. Therefore, B * C * A. The proof of the converse issimilar.ВА, АССА, and CBBC, it is alsoAny function that has the properties spelled out in Theorem 3.2.7 can be used tomeasure distances. Such a function is called a metric.Definition 3.2.9. A metric is a function D : P × P→ R such that1. D(P, Q)2. D(P, Q) > 0 for every P and Q, and3. D( P, Q) = 0 if and only if P = Q.D(Q, P) for every P and Q,A version of the triangle inequalityD(P, Q) < D(P, R) + D(R, Q)is often included as part of the definition of metric, but we do not include it in thedefinition because we will prove the triangle inequality as a theorem in the next chapter.'The familiar distance formula from calculus and high school geometry is an exampleof a metric.EXAMPLE 3.2.10The Euclidean metricDefine the distance between points (x1, yı) and (x2, y2) in the Cartesian plane byd((x1, y1), (x2, y2)) = J(x2x1)² + (y2y1 )2.This metric is called the Euclidean metric. The verification that d satisfies the conditionsin the definition of metric is left as an exercise (Exercise 2).There are other ways in which to measure distances in the Cartesian plane. Here isan example that will be important later in the chapter.EXAMPLE 3.2.11The taxicab metricDefine the distance between points (x1, y1) and (x2, y2) in the Cartesian plane byp((x1, y1), (x2, y2)) = |x2x1| + |y2 – yil.This metric is called the taxicab metric. The verification that p satisfies the conditions inthe definition of metric is left as an exercise (Exercise 3).The name taxicab is given to the second metric because the distance from (x1, y1)to (x2, y2) is measured by traveling along a line parallel to the x-axis and then along oneparallel to the y-axis in much the same way thatrectangular grid of streets and cannot travel directly from point A to point B. There isalso a natural way to measure distances between points on the sphere.taxicab in Manhattan must follow aWhat we are calling a metric should more properly be called a semimetric. However, all theexamples of metrics we will consider satisfy the triangle inequality and are metrics according tothe standard definition. We omit the triangle inequality from the definition only because it is anunnecessary assumption in this context.40Chapter 3Axioms for Plane Geometry(X, ) 10:57Gerard A. Venema Foundations of Geometry 2011PDF - 3 MBFIGURE 3.7: Two circles in the rational planecoordinate functions use real numbers will allow us to fill that gap. Specifically, we willprove that if one circle contains a point that is inside a second circle and another pointthat is outside the second circle, then the two circles must intersect. Even using thepowerful axioms in this chapter the proof is not easy, however, and we will not attemptthat particular proof until Chapter 8.XERCISES 3.21. Prove: If l and m are two lines, the number of points in l n m is either 0, 1, or o.2. Show that the Euclidean metric defined in Example 3.2.10 is a metric (i.e., verify thatthe function d satisfies the three conditions in the definition of metric on page 39).3. Show that the taxicab metric defined in Example 3.2.11 is a metric (i.e., verify that thefunction p satisfies the three conditions in the definition of metric on page 39).4. Show that the spherical metric defined in Example 3.2.12 is a metric (i.e., verify that thefunction s satisfies the three conditions in the definition of metric on page 39).5. Betweenness in taxicab geometry. Let A =(0, 0), В %3D (1,0) and C %3D(1, 1) and let pdenote the taxicab metric.(a) Find all points P such that p(A, P) + p(P, B) = p(A, B). Draw a sketch in theCartesian plane.(b) Find all points P such that p( A, P) + p( P, C) = p(A, C). Draw a sketch in theCartesian plane.6. Betweenness on the sphere. Let A, B, and C be points on the sphere S². Define C to bebetween A and B if A, B, and C are collinear and s(A, C) + s(C, B) = s(A, B), wheres is the metric defined in Example 3.2.12. Also define segment in the usual way.(a) Find all points that are between A and C in case A and C are nonantipodal points.Sketch the segment from A to C.(b) Find all points that are between A and C in case A and C are antipodal points.Sketch the segment from A to C.7. Find all points (x, y) in R² such that p( (0, 0), (x, y)) = 1, where p is the taxicab metric.Draw a sketch in the Cartesian plane. (This shape might be called a “circle" in thetaxicab metric.)8. The square metric. Define the distance between two points (x1, y1) and (x2, y2) in R²by D((x1, yı ), (x2, y2)) = max{|x2(a) Verify that D is a metric.(b) Find all points (x, y) in R² such that D((0,0), (x, y))Cartesian plane. (This should explain the name square metric.)x1), ly2yı|}.4= 1. Draw a sketch in the4max{a, b} denotes the larger of the two real numbers a and b.46 Chapter 3Axioms for Plane Geometry9. Verify that the functions defined in Example 3.2.14 are coordinate functions.10. Verify that the functions defined in Example 3.2.15 are coordinate functions.11. Let l be a line in the Cartesian plane R2. Find a function f :l → R that is a coordinatefunction for l in the square metric. (See Exercise 8 for definition of square metric.)12. Assume that f :l → R is a coordinate function for le.(a) Prove that – f is also a coordinate function for l.(b) Prove that g : l → R defined by g( P) = f(P) + c for some constant c is also acoordinate function for l.(c) Prove that if h : l → R is any coordinate function for l then there must exist aconstant c such that either h(P) = f(P) + c or h( P) = -f(P) + c.13. Let l be a line and let f : l → R be a function such that PQ = \f(P)every P, Q E l. Prove that f is a coordinate function for l.14. Prove the following fact from high school algebra that was needed in theTheorem 3.2.17: If x and y are two nonzero real numbers such that |x| + |y|then either both x and y are positive or both x and y are negative.15. Prove Corollary 3.2.20.16. Prove that if C E AB and C # A, then AB =17. Prove existence and uniqueness of midpoints (Theorem 3.2.22).AĆ.

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the function d satisfies the three condition in the definitión of metric on page 39). 3. Show that the taxicab metric defined in Example 3.2.11 is a metric (i.e., verify that the function p satisfies the three conditions in the definition of metric on page 39). Show thot the snhorico1motric deined in Evomnple 3 2 12 is o metrie vorify that the