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3. Consider a geometry with five points a, b, c, d, and e. Let the lines...
3) Let us build a geometry, S, using the three axioms of incidence geometry with one additional a...
3) Let us build a geometry, S, using the three axioms of incidence geometry with one additional axiom added: Incidence Axiom l: For every point P and every point Q (P and Q not equal), there exists a unique line, I, incident with P and Q. Incidence Axiom 2: For every line / there exist at least two distinct points incident with Incidence Axiom3: There exist (at least) three distinct points with the property that no line is incident with...
3) Let us build a geometry, S, using the three axioms of incidence geometry with one...
3) Let us build a geometry, S, using the three axioms of incidence geometry with one additional axiom added: Incidence Axiom l: For every point P and every point Q (P and Q not equal), there exists a unique line, I, incident with P and Q. Incidence Axiom 2: For every line / there exist at least two distinct points incident with Incidence Axiom3: There exist (at least) three distinct points with the property that no line is incident with...
Exercise 3. Let G be a model of incidence geometry in which every line contains at...
Exercise 3. Let G be a model of incidence geometry in which every line contains at least three distinct points. (i) Prove that if I and m are distinct lines, then there erists a point P such that P does not lie on l or m. (ii) Prove that if G additionally satisfies the Elliptic Parallel Postulate and G has a finite number of points, then every line contains the same number of points.
Part II. (4 pts) Given the axiom set for the Incidence Geometry as below: Undefined terms: point, line, on Definitions: 1. Two lines are intersecting if there is a point on both. 2. Two lines are...
Part II. (4 pts) Given the axiom set for the Incidence Geometry as below: Undefined terms: point, line, on Definitions: 1. Two lines are intersecting if there is a point on both. 2. Two lines are parallel if they have no point in common. Axioms: I. Given any two distinct points, there is a unique line on both. II. Each line has at least two distinct points on it. III. There exist at least three points. IV. Not all points...
2) (i) State the converse of the Alternate Interior Angle Theorem in Neutral Geometry. (ii) Prove...
2) (i) State the converse of the Alternate Interior Angle Theorem in Neutral Geometry. (ii) Prove that if the converse of the Alternate Interior Angle Theorem is true, then all triangles have zero defect. [Hint: For an arbitrary triangle, ABC, draw a line through C parallel to side AB. Justify why you can do this.] 5) Consider the following statements: I: If two triangles are congruent, then they have equal defect. II: If two triangles are similar, then they have...
(1) Assume the axioms of metric geometry. Let A, B, C, D be distinct collinear points....
(1) Assume the axioms of metric geometry. Let A, B, C, D be
distinct collinear points. Let f : l → R be a coordinate function
for the line l that crosses all of A, B, C, D. Suppose f(A) <
f(B) < f(C) < f(D). Prove that AD = AB ∪ BC ∪ CD. (2) Assume
the axioms of metric geometry. Let A, B, C, D be distinct collinear
points. Suppose A ∗ B ∗ C and B ∗...
8. True or false (in absolute geometry unless otherwise stated.) (a) If A and D are...
8. True or false (in absolute geometry unless otherwise stated.) (a) If A and D are points on opposite sides of BC and LABC BCD, then AB II CD (b) If two lines are parallel, then they are equidistant from each other. (c) If oABCD is a quadrilateral with right angles at A, B, and C, then LD is also a right angle. (d) Euclid's Parallel Postulate is equivalent to the following statement: Every point in the interior of an...
1) Assume that (P. L. d) satisfies postulates 1-6 of neutral geometry. Let C P be...
1) Assume that (P. L. d) satisfies postulates 1-6 of neutral geometry. Let C P be a ne. We denote by H and H the two sides of (see postulate 6) (a) Show that both sides H+ and H are non-empty. (20 points) (b) Show that both sides H+ and H are infinite sets. (10 points)
p. 238 2. Verify the claim made in Example 6.5 that there is only one line...
p. 238 2. Verify the claim made in Example 6.5 that there is only one line parallel to the given line through a point not on the line. allel lines are similar to those in Euclidean geometry. Example 6.5 Let S = {a,b,c,d). Let the lines in this geometry be the sets (a, b), la, c), fa, d), b, d, b, d), and (c,d). This geometry satisfies I(1)-I(3), as you will verify in the Student Learning Opportunities. But in this...
Let a, b, c, d and e be the first five digits of your La Trobe...
Let a, b, c, d and e be the first five digits of your La Trobe student number. i.e. if your student number is 12345678 then a = 1,b=2,c=3, d = 4 and e = 5. Use your appropriate values in the question below. 2 a Consider the line. L = {(2,3, 2): ?- } y-b 3 2 (a) Write L in parametric form. (b)0 Find the coordinates of the point P at which L intersects the plane z =...
3) Let us build a geometry, S, using the three axioms of incidence geometry with one additional axiom added: Incidence Axiom l: For every point P and every point Q (P and Q not equal), there exists a unique line, I, incident with P and Q. Incidence Axiom 2: For every line / there exist at least two distinct points incident with Incidence Axiom3: There exist (at least) three distinct points with the property that no line is incident with...
3) Let us build a geometry, S, using the three axioms of incidence geometry with one additional axiom added: Incidence Axiom l: For every point P and every point Q (P and Q not equal), there exists a unique line, I, incident with P and Q. Incidence Axiom 2: For every line / there exist at least two distinct points incident with Incidence Axiom3: There exist (at least) three distinct points with the property that no line is incident with...
Exercise 3. Let G be a model of incidence geometry in which every line contains at least three distinct points. (i) Prove that if I and m are distinct lines, then there erists a point P such that P does not lie on l or m. (ii) Prove that if G additionally satisfies the Elliptic Parallel Postulate and G has a finite number of points, then every line contains the same number of points.
Part II. (4 pts) Given the axiom set for the Incidence Geometry as below: Undefined terms: point, line, on Definitions: 1. Two lines are intersecting if there is a point on both. 2. Two lines are parallel if they have no point in common. Axioms: I. Given any two distinct points, there is a unique line on both. II. Each line has at least two distinct points on it. III. There exist at least three points. IV. Not all points...
(1) Assume the axioms of metric geometry. Let A, B, C, D be
distinct collinear points. Let f : l → R be a coordinate function
for the line l that crosses all of A, B, C, D. Suppose f(A) <
f(B) < f(C) < f(D). Prove that AD = AB ∪ BC ∪ CD. (2) Assume
the axioms of metric geometry. Let A, B, C, D be distinct collinear
points. Suppose A ∗ B ∗ C and B ∗...
8. True or false (in absolute geometry unless otherwise stated.) (a) If A and D are points on opposite sides of BC and LABC BCD, then AB II CD (b) If two lines are parallel, then they are equidistant from each other. (c) If oABCD is a quadrilateral with right angles at A, B, and C, then LD is also a right angle. (d) Euclid's Parallel Postulate is equivalent to the following statement: Every point in the interior of an...
1) Assume that (P. L. d) satisfies postulates 1-6 of neutral geometry. Let C P be a ne. We denote by H and H the two sides of (see postulate 6) (a) Show that both sides H+ and H are non-empty. (20 points) (b) Show that both sides H+ and H are infinite sets. (10 points)
p. 238 2. Verify the claim made in Example 6.5 that there is only one line parallel to the given line through a point not on the line. allel lines are similar to those in Euclidean geometry. Example 6.5 Let S = {a,b,c,d). Let the lines in this geometry be the sets (a, b), la, c), fa, d), b, d, b, d), and (c,d). This geometry satisfies I(1)-I(3), as you will verify in the Student Learning Opportunities. But in this...
Let a, b, c, d and e be the first five digits of your La Trobe student number. i.e. if your student number is 12345678 then a = 1,b=2,c=3, d = 4 and e = 5. Use your appropriate values in the question below. 2 a Consider the line. L = {(2,3, 2): ?- } y-b 3 2 (a) Write L in parametric form. (b)0 Find the coordinates of the point P at which L intersects the plane z =...
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