3.2 #5 (0,0)(2,0)FIGURE 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.EXERCISES 3.21. Prove: If l and m are two lines, the number of points in l n m is either 0, 1, or oo.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))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, y1), (x2, y2))(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.)%3D1, where p is the taxicab metric.max{\x2x1], [y2- yıl}.4= 1. Draw a sketch in the4max{a, b} denotes the larger of the two real numbers a and b.46Chapter 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 l.(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) + cor h(P) = – f(P) + c.13. Let l be a line and let f : e → R be a function such that PQ = |f(P) – f(Q)| forevery 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 the proof ofTheorem 3.2.17: If x and y are two nonzero real numbers such that |x| + ]y[ = ]x + yl,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).18. (Segment Construction Theorem) Prove the following theorem. If AB is a segment andAČ.

Question

3.2 #5 (0,0)(2,0)FIGURE 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.EXERCISES 3.21. Prove: If l and m are two lines, the number of points in l n m is either 0, 1, or oo.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))Draw a sketch in the Cartesian plane. (This shape might be called a &#34;circle&#34; in thetaxicab metric.)8. The square metric. Define the distance between two points (x1, y1) and (x2, y2) in R²by D((x1, y1), (x2, y2))(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.)%3D1, where p is the taxicab metric.max{\x2x1], [y2- yıl}.4= 1. Draw a sketch in the4max{a, b} denotes the larger of the two real numbers a and b.46Chapter 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 l.(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) + cor h(P) = – f(P) + c.13. Let l be a line and let f : e → R be a function such that PQ = |f(P) – f(Q)| forevery 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 the proof ofTheorem 3.2.17: If x and y are two nonzero real numbers such that |x| + ]y[ = ]x + yl,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).18. (Segment Construction Theorem) Prove the following theorem. If AB is a segment andAČ.

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