Homework

Math 306, Spring 2002

Part I

1. Enumerate those propositions in Euclid Book I which are dependent on Postulate 5 (the Parallel Postulate).
2. Greenberg I.Ex.12: What is the flaw in the proof that all triangles are isosceles?
3. Greenberg I.MajEx.4: The number p=(1+√5)/2 was called the golden ratio by the Greeks, and a rectangle whose sides are in this ratio is called a golden rectangle. Prove that a golden rectangle can be constructed with straightedge and compass as follows:
   1. Construct a square ABCD.
   2. Construct midpoint M of AB.
   3. Construct point E such that B is between A and E and MC ≅ ME.
   4. Construct the foot F of the perpendicular from E to DC.
   5. Then rectangle AEFD is a golden rectangle (use the Pythagorean theorem for triangle MBC).
   6. Moreover, rectangle BEFC is another golden rectangle (first show that 1/p=p-1).
4. Greenberg II.MajEx.5: Consider the following interpretation of incidence geometry. Begin with a punctured sphere in Euclidean three-space, i.e., a sphere with one point N removed. Interpret "points" as points on the punctured sphere. For each circle on the original sphere passing through N, interpret the punctured circle obtained by removing N as a "line". Interpret "incidence" in the Euclidean sense of a point lying on a punctured circle. Is this interpretation a model of incidence geometry? If so, what parallel property does it have? Is it isomorphic to any other model you know? (Hint: If N is the north pole project the punctured sphere from N onto the plane Π tangent to the sphere at the south pole. Use the fact that planes through N cut out circles on the sphere and lines in Π.
5. Greenberg II.MajEx.10: A "point" [x, y, z] in the real projective plane is determined by an ordered triple (x, y, z) of real numbers that are not all zero, and it consists of all the ordered triples of the form (kx, ky, kz) for all real numbers k≠0; thus [kx, ky, kz] = [x, y, z]. A "line" in the real projective plane is determined by an ordered triple (u, v, w) of real numbers that are not all zero, and it is defined as the set of all "points" [x, y, z] whose coordinates satisfy the linear equation ux+vy+wz=0. "Incidence" is defined as set membership. Verify that all the axioms for a projective plane are satisfied by this interpretation. Prove that by taking z=0 as the equation of the "line at infinity", by assigning the affine "point" (x, y) the "homogeneous coordinates" [x, y, 1], and by assigning affine "lines" to projective "lines" in the obvious way, the real projective plane becomes isomorphic to the projective completion of the real affine plane. Prove that the following models are isomorphic to the real projective plane:
   1. Fix a distinguished point O in three-space. "Points" are lines passing through O, "lines" are planes containing O, and "incidence" is the usual relation of a line lying in a plane.
   2. Fix a sphere in Euclidean three-space. Two points on the sphere are called antipodal if they line on a diameter of the sphere; e.g., the north and south poles are antipodal. Interpret a "point" to be a set {P,P'} consisting of two antipodal points on the sphere. Interpret a "line" to be a great circle C on the sphere. Interpret a "point" {P,P'} to "lie on" a "line" C if one of the points P, P' lies on the great circle C (in which event the other point also lies on C).
6. Greenberg III.Ex.12: Prove the Crossbar Theorem:
   If ray AD is between ray AC and ray AB, then ray AD intersects segment BC.
7. Greenberg III.Ex.28: Prove (using the Hilbert Axioms) that an equiangular triangle (all angles congruent to one another) is equilateral.
8. Greenberg IV.Ex.9: The following purports to be a proof in neutral geometry of the SAA criterion for congruence. Find the flaw.
   Given AC ≅ DF, ∠A ≅ ∠D, ∠B ≅ ∠E. Then ∠C ≅ ∠F, since (∠C)° = 180°-(∠A)° - (∠B)° = 180° - (∠D)° - (∠E)° = (∠F)°. Hence, triangle ABC ≅ triangle DEF by ASA.
   
9. Greenberg IV.Ex.10: Justify each step of the following proof of the SAA congruence criterion under the Neutral Geometry hypotheses:
   1. Assume side AB is not congruent to side DE.
   2. Then AB ≤ DE or DE ≤ AB.
   3. If DE ≤ AB, then there is a point G between A and B such that AG≅DE.
   4. Then triangle CAG≅ triangle FDE.
   5. Hence ∠AGC≅∠E.
   6. It follows that ∠AGC≅∠B.
   7. This contradicts a certain theorem (identify the theorem)
   8. Therefore, DE is not less than AB.
   9. By a similar argument involving a point H between D and E, AB is not less than DE.
   10. Therefore, triangle ABC≅ triangle DEF.

Part IIA

1. Chap 4, Major Exercise 4 (Converse to Triangle Inequality)
2. Chap 5, Exercise 8 (Legendre Proof of Parallel Post)
3. Chap 5, Exercise 2
4. Chap 6, Exercise 2
5. Chap 6, Exercise 5 (Triangle Defect)

Part IIB

1. Consider the following four axiom systems for geometries:
   • I: Incidence Geometry (3 axioms on pgs 50-51 of Greenberg)
   • E: Euclidean Geometry (Hilbert Axioms)
   • N: Neutral Geometry (Hilbert Axioms without Parallel Postulate)
   • H: Hyperbolic Geometry (Hilbert's Axioms, replacing the Parallel Axiom with the Hyperbolic Axiom)
   For each of the following statements (a-e) state which one of the following cases (i-iii) holds. Add a short justification (e.g. "This is Hilbert Axiom X" or "A counterexample is Blah"):
   1. The statement can be proven
   2. The negation of the statement can be proven
   3. Neither can be proven
   Statements:
   1. Given a line l and a point P not on l there is at least one line through P parallel to l.
   2. Given a line l and a point P not on l there are an infinite number of lines through P, parallel to l.
   3. Every line has at least two points.
   4. Every line has an infinite number of points.
   5. If two lines, l and m, are parallel, then all points of m are are on the same side of l. (see defn of "same side", Greenberg, pg 76)
   [Problem 3 from the mid-term, with some clarifications]
2. Symmetry of Limiting Parallelism. If ray BA is a limiting parallel for ray CD (written "BA|CD"), then ray CD is a limiting parallel for ray AB. Justify the unjustified steps in the proof:

   Proof:

   1. Assume the negation, that CD is not a limiting parallel for AB.
   2. Then some interior ray CE does not intersect BA
   3. Point E, which so far is just a label, can be chosen so that ∠BEC < ∠ECD, by the important corollary to Aristotle's axiom, Chapter 3.
   4. Segment BE does not intersect ray CD.
   5. Interior ray BE intersects ray CD in a point F, and B*E*F.
   6. Since ∠BEC is an exterior angle for triangle EFC, ∠BEC > ∠ECF.
   7. Contradiction.

   [Greenberg Chap 6, MajEx 2]
3. Given any angle ∠A'OA. It is a theorem in hyperbolic geometry that there is a unique line l called the line of enclosure of this angle such that l is limiting parallel to both sides, ray OA' and ray OA. Only the idea of the proof is given here. Fill in the details:

   Assume that A and A' are chosen so that OA≅OA'. Let A'Ω be the limiting parallel ray to ray OA through A', and AΣ the limiting parallel ray to ray OA' through A. Let the rays r and r' be the bisectors of ∠ΣAΩ and ∠ΣA'Ω, respectively. The idea of the proof is to show that the lines m and m' containing these rays are neither intersecting nor asymptotically parallel, so that, by Thm 6.7, they have a unique common perpendicular l that turns out to be the line of enclosure of ∠A'OA.

   [Greenberg Chap 6, MajEx 8]
4. Let l be a Poincare line that is not a diameter of γ, l is then an arc of a circle δ orthogonal to γ. Prove that hyperbolic reflection across l is represented in the Poincare model by inversion in δ. (Hint: Use Proposition 7.10 and the corollary to Prop 7.6)
   [Greenberg Chap 7, Ex P-4]
5. In a Euclidean plane, prove that triangle ABC is similar to triangle A'B'C' if and only if there is a similarity sending A, B, C respectively onto A', B', C', and that similarity is unique. (Hint: Use Lemma 9.2, Prop 9.5, and Ex 20 of Chap 5.)
   [Greenberg Chap 9, Ex 6]