Math 348 (2006): Assignment #1
due Monday, July 10, 2006
1. Let ABC be a triangle. Suppose that B C. Prove that AB = = AC, so the triangle is isosceles. Hint: Mimic the proof of Pappuss Theorem but use ASA instead of SAS. 2. Show that a rectangle is a squ
MATH 348: Solved Emergency Problems, Week 4
Leonid Chindelevitch
August 26, 2008
1
Monday, July 28, 2008
1. Problem: From each of the centers O1 , O2 of two circles 1 , 2 tangents
are drawn to the other circle. Prove that equal chords are intercepted on
t
MATH 348: Solved Emergency Problems, Week 3
Leonid Chindelevitch
August 24, 2008
1
Monday, July 21, 2008
1. Problem: Let ABC be an equilateral triangle inscribed in a circle , and
let P be a point on , with A the vertex farthest away from P . Show that
d(
MATH 348: Solved Emergency Problems, Week 2
Leonid Chindelevitch
August 25, 2008
1
Monday, July 14, 2008
1. Problem: ABCD is a cyclic quadrilateral with extended sides AD, BC
meeting at Q and extended sides BA, CD meeting at P . Prove that the
quadrilater
MATH 348: Solved Emergency Problems, Week 1
Leonid Chindelevitch
July 13, 2008
1
Tuesday, July 8, 2008
1. Problem: Given a circle , construct its center O using only a compass.
Solution: Let r be the (unknown) radius of . Take a point A on ,
and draw a ci
MATH 348: Assignment 5 (nal version)
Leonid Chindelevitch
July 29, 2008
1. Suppose that A, B, C, D are 4 distinct points in the plane, with AB CD
and AC BD. Prove that AD BC . Hint: Consider several cases!
2. In an isoceles triangle, prove that the sum of
MATH 348: Assignment 4 (nal version)
Leonid Chindelevitch
August 2, 2008
1. Two circles, 1 and 2 , and a line l are given. Locate a line, parallel to
l, so that the distance between the points at which this line intersects 1
and 2 is equal to the length o
MATH 348: Assignment 3 (nal version)
Leonid Chindelevitch
July 22, 2008
1. Show that if a regular n-gon can be constructed with a ruler and compass,
then a regular 2n-gon can also be constructed with a ruler and compass.
Hint: Use trigonometry and half-an
MATH 348: Assignment 2 (nal version)
Leonid Chindelevitch
July 15, 2008
1. Let the triangle ABC have circumradius R. Prove that its area is given
by
abc
.
S=
4R
Comment: This is an ordinary proof, so you may use the fact sheet.
2. Prove that the following
MATH 348: Assignment 1 (nal version)
Leonid Chindelevitch
July 11, 2008
1. Choose a geometer you are interested in and write a brief biography explaining their contribution to the geometry of their time.
Thales
Pythagoras
Euclid
Khayyam Archimedes Liu Hui
Math 348 (2006): Assignment #4: Solutions
1. We know that there are three types of orientation-reversing isometries of E 3 : reflection, glide, and rotatary reflection. We also know that a half-turn H about a line can be written as the composition rM2 rM1
Math 348 (2006): Assignment #4
due Monday, July 31, 2006
1. Show that any orientation reversing isometry of E 3 can be written as the composition of a reection and a half-turn. 2. Consider a dilative rotation in E 3 . This is the composition of a dilation
Math 348 (2006): Assignment #3: Solutions
1. Let be a circle passing through O. Let A be the point on diametrically opposite from O. Let A be the inverse of A. Now consider any point P on other than O and A. Since OP A is an angle inscribed in a semi-circ
Math 348 (2006): Assignment #3
due Monday, July 24, 2006
For problems 1, 2, and 3, let be a circle with centre O and radius k. 1. Show directly, using the denition of inversion, that if is a circle passing through O, then its inverse (with respect to ) is
Math 348 (2006): Assignment #2: Solutions
1. Let and be a pair of parallel lines. Suppose the line m intersects and at points P and P , respectively. Since , the alternate interior angles are congruent. Now let f be an isometry. We know it takes lines to
Math 348 (2006): Assignment #2
due Monday, July 17, 2006
1. Let f be an isometry of the plane. Show that f takes a pair of parallel lines to a pair of parallel lines. (You can use the fact that we know it takes lines to lines, and that it preserves angles
Math 348 (2006): Assignment #1: Solutions
1. Consider the correspondence of vertices A A, B C, C B. Under this correspondence, we have that B C (by hypothesis) and also C B. Finally, = = the sides contained between these corresponding pairs of vertices ar
GEOMETRY #1
DYLAN CANT
Problem 1. Let A, B, C, D be distinct points in the plane and suppose that the lines AC
and BD intersect in a point E. Prove that the following two conditions are equivalent:
(i) |AB| = |BC| = |CD| = |DA|,
(ii) |AE| = |EC|, |BE| = |