## MATH 371 Assignments Spring, 2003

TEXTS: Fundamentals of Geometry by B. Meserve and J. Izzo, A.W. (1969) - ON LINE with  HSU ONCORE
The Elements by Euclid, 3 volumes, edited by T.L. Heath, Dover (1926)
Proof in Geometry by A.I Fetisov, Mir (1978)
Here's Looking at Euclid..., by J.Petit, Kaufmann (1985).
Flatland By E. Abbott, Dover.

 Week Monday Wednesday Reading for the week. Problems Due on Wednesday of the next week Friday 1 1/20 No Class 1/23 1.1 Beginnings  What is Geometry?  The Pythagorean Theorem 1/25Intro to Geometer's Sketchpad/ Wingeom    Transformations M&I:1.1, 1.2  E:I Def'ns, etc. p153-5;  Prop. 1-12,22,23,47  A:.Complete in three weeks M&I p5:1-8,11 Due: 1/29 2 1/27 The Pythagorean Theorem 1/29  1.2 Equidecomposable Polygons 2/31 M&I 1.2, 1.3  E: I Prop. 16, 27-32, 35-45. M&I: p10:1,2,5,10,11-13 Due:2/6  Prove:The line segment connecting the midpoints of  two sides of a triangle is parallel to the third side and is congruent to one half of the third side.  Lab Exercises 1: Due: 2/31  Construct a sketch with technology of  1. Euclid's Proposition 1 in Book I.  2. Euclid's Proposition 2 in Book I.  3. One "proof" of the Pythagorean Theorem. 3 2/4 More on Constructions  Isometries  1.1 Def'ns- Objects  1.2 Constructions 2/6 1.3 Geometry and numbers 2/8 1.4 Continuity M&I 1.3,1.4  E: III Prop. 1-3, 14-18, 20, 21, 10  F. Sect. 11, 25, 31 M&I: p17:5, 8-11  p11: 16-19, 24, *27  Due:2/13  Problem  Set 1 Due:2/13  Lab Exercises 2: Due by 2/8.  Do Construction 3, 4, 6, 7, and 8 from Meserve and Izzo Section 1.2.  BONUS:Show how to "add" two arbitrary triangles to create a single parallelogram. 4 2/11 Transformations - Isometries 2/13 Coordinates and Transformations 2/15 Inversion  Begin Affine Geometry M&I:1.5, 1.6, 2.1   E: V def'ns 1-7;VI: prop 1&2  F. Sect. 32 M&I:1.6:1-12,17,18 Due 2/20  Problem Set 2 Due 2/20  Lab Exercises 3: Due 2/15.   1. Construct a scalene triangle using Wingeometry. Illustrate how to do i) a translation by a given "vector", ii) a rotation by a given angle measure, and iii) reflections across a given line..  2. Create a sketch that shows that the product of two reflections is either a translation or a rotation. 5 2/18 Affine Geometry 2/20 Affine Geometry 2/22 More affine geometry. Orthogonal circles and Inversion. M&I: 2.1, 2.2,  E:IV Prop. 3-5 M&I: p23: 9,10 (analytic proofs) Due 2/27  Lab Exercises 4: Due 2/22.  1. Draw a figure showing the product of three planar reflections as a glide reflection.  2. Draw a figure illustrating the effects of a central similarity on a triangle using magnification or dilation that is a) positive number >1, b) a positive number <1, and c) a negative number. 6 2/25 More Inversion and Affine Geometry (planar coordinates) 2/27 Breath 3/1Homogeneous Coordinates. M&I: 2.1,2.2 Problem Set 3 (Isos Tri) Due  [4 Points for every distinct correct proof of any of these problems.]  Lab Exercises 5: Due 3/1.  Construct the inverse of a point with respect to a circle a) when the point is inside the circle; b) when the point is outside the circle. 7 3/4 Continuation on coordinates. 3/6 Begin Synthetic Geometry [Finite] 3/8 More on Synthetic geometry. M&I: 3.1,3.2, 3.5 Due   M&I: 3.5: 1,3,4,5,10,11  Lab Exercise 6: Due 3/8 . See Notes [Abridged]  1. Draw two intersecting circles O and O' and measure the angle between them.  2. Given a circle O and two interior points A and B, construct an orthogonal circle O' through A and B.  3.Use inversion with respect to the circle OP to invert

Problem Set 1

DEFINITIONS: A figure C is called convex if for any two points in the figure, the line segment determined by those two points is also contained in the figure.
That is, if A is a point of C and B is a point of C then the line segment AB is a subset of C.

If F and G are figures then F int G  is { X : X in F and X  in G }.
F int G is called the intersection of F and G.
If A is a family of figures (possibly infinite), then  int A = { X : for every figure F in the family A, X  is in F }.
int A is called the intersection of the family A.
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1. Prove: If F and G are convex figures , then F int G is a convex figure.

2. Give a counterexample for the converse of problem 1.

3. Prove: If A is a family of convex figures, then int A is a convex figure.

4. Prove: The line segment RS is convex. [ Refer to M & I pg.2.]

Problem Set 2

1. Suppose n is a natural number. Given P0 and P1 , prove by induction that you can construct with straight edge and compass (SEC) a point P sqrt(n)   which will correspond to the number  sqrt(n) on a Euclidean line.

2. Suppose we are given P0, P1, and P a where P a corresponds to the real number a>0. Give a construction with SEC of a point Psqrt(a) which will correspond to the number  sqrt(a) on a Euclidean line.

3. Given points P0, P1, Px, and Py on a Euclidean line corresponding to the real numbers x>0 and y>0, give constructions with SEC for the following points.

 a) P x + y b) P x - y c) P x *y d) P 1/x
4. Construct with SEC on a Euclidean line:  sqrt( sqrt(5)/sqrt(3)  + sqrt(6) ).

5. Suppose that d(A,B) = d(A',B') and that l is the perpendicular bisector of the line segment  AA'. Let B'' be the reflection of B across l, i.e., B''= Tl(B). Prove that if  B' is not equal to B''  then A' lies on the perpendicular bisector of the line segment.

Problem Set 3

1.  Prove: Two of the medians of an isosceles triangle are congruent.

2.  Prove: If two of the medians of a triangle are congruent then the triangle is isosceles.

3.  Prove: The angle bisectors of congruent angles of an isosceles triangle are congruent.

4.  Prove:  If two of the angle bisectors of a triangle are congruent then the triangle is isosceles.

Problem Set  4

1. Use an affine line with P0 , P1 , and Pinf given. Show a construction for P1/2 and P2/3.

2. Use an affine line with P0 , P1 , and Pinf   given. Suppose x > 1.
Show a construction for  Px2 and Px3 when Px is known.

3. D is a circle with center N tangent to a line l at the point O and C is a circle that passes through the N and is tangent to l at O as well.
Suppose P is on l and PN intersect C = {Q}; Q' is on C so that Q'Q is parallel to ON; and {P'} = NQ'  intersect l. Prove: a) P and Q are inverses with respect to the circle D.
b) P' and Q' are inverses with respect to the circle D.
c) P and P' are inverses with respect to the circle with center at O and radius ON.

4. Suppose C is a circle with center O and D is a circle with O  an element of D.
Let I be the inversion transformation with respect to C.

Prove: There is a line l, where I(P) is an element of  l for all P  that are elements of  D -{O}.