Mathematics 226H Geometry I: Euclidean geometry
Trent University, Winter 2008
Solutions to Problem Set #6
1. (Exercise 2C.3) Given ABC, draw line W V through A parallel to BC, line U W
through B parallel to AC, and line U V through C parallel to AB. Show
Mathematics 226H Geometry I: Euclidean geometry
Trent University, Winter 2008
Solutions to Problem Set #10
1. (Exercise 4A.2) Let U and V be points on sides AB and AC, respectively, of ABC
and suppose that U V is parallel to BC. Show that the intersection
Mathematics 2260H Geometry I: Euclidean geometry
Trent University, Winter 2012
Solutions to Assignment #5
Tinkering with triangles
In both of the questions below suppose D and E are the midpoints of sides AB and
AC, respectively, of ABC.
1. Show that DE
B
Mathematics 2260H Geometry I: Euclidean geometry
Trent University, Winter 2012
Solution to Assignment #4
Similarity
Euclids Elements doesnt get into similarity until Book VI, but its a concept that
comes in very handy for some things, a few of which we wi
Mathematics 2260H Geometry I: Euclidean geometry
Trent University, Winter 2012
Solutions to Assignment #1
A geometry on a paraboloid
We will dene a geometry G of points and lines on (the surface of) the paraboloid
z = x2 + y 2 1, part of which pictured be
Mathematics 226H Geometry I: Euclidean geometry
Trent University, Fall 2006
Solutions to Problem Set #1
Please note that these solutions do not include any diagrams. To make sure you
understand how they work, draw the diagrams for yourself, especially for
Mathematics 226H Geometry I: Euclidean geometry
Trent University, Winter 2008
Solutions to Problem Set #2
1. (Exercise 1B.7) Suppose BY and CZ are altitudes of ABC and |BY | = |CZ|. Show
that |AB| = |AC|. [5]
Solution 1. Since we have |BY | = |CZ| (given)
Mathematics 226H Geometry I: Euclidean geometry
Trent University, Fall 2006
Take-Home Final Examination
Due on Friday, 22 December, 2006.
Instructions: Do all three of parts A C, and, if you wish, part
as well. Show all
your work. You may use your textboo
Mathematics 226H Geometry I: Euclidean geometry
Trent University, Fall 2006
Solutions to Problem Set #3
1. (Exercise 1H.2) Show that the three medians of a triangle go through a common point
[the centroid of the triangle]. [5]
Hint: Use Problem 1.30.
Solu
Mathematics 226H Geometry I: Euclidean geometry
Trent University, Winter 2008
Solutions to Problem Set #1
1. Go through Euclids proof of Proposition I-1 in the Elements and identify at each step
the use, implicit or explicit, of his denitions, postulates,
Mathematics 226H Geometry I: Euclidean geometry
Trent University, Fall 2006
Solutions to Problem Set #2
1. (Exercise 1E.1) Draw two medians of a triangle. This subdivides the interior of the
triangle into four pieces: three triangles and a quadrilateral.
Mathematics 226H Geometry I: Euclidean geometry
Trent University, Fall 2006
Problem Set #5
1. (Exercise 2A.3) Given acute angled ABC, extend the altitudes from A, B, and C
to meet the circumcircle at points X, Y , and Z, respectively. Show that lines AX,
Mathematics 226H Geometry I: Euclidean geometry
Trent University, Fall 2006
Solutions to Problem Set #4
1. (Exercise 1H.6) In the gure below [Figure 1.45 in the text], line segments P A, P B,
and P C join point P to the three vertices of ABC. We have chos
Mathematics 226H Geometry I: Euclidean geometry
Trent University, Fall 2006
Solution to Problem Set #6
1. Suppose ABC is not equilateral. Must the incentre of this triangle be on its Euler
line? Prove it or give a counterexample. [10]
Solution. The incent
Mathematics 226H Geometry I: Euclidean geometry
Trent University, Fall 2006
Solutions to Problem Set #7
1. (Exercise 2E.2) Suppose that the centroid and incentre of ABC are the same point.
Prove that the triangle is equilateral. [5]
Solution. Suppose AP ,
Mathematics 2260H Geometry I: Euclidean geometry
Trent University, Winter 2012
Solutions to Assignment #6
Cevas Theorem
The following result appears to have been rst obtained by the Arab mathematician
Yusuf ibn Ahmad al-Mutaman ibn Hud, who also served as
Mathematics 2260H Geometry I: Euclidean geometry
Trent University, Winter 2013
Take-Home Final Examination
Due on Friday, 19 April, 2013.
Instructions: Do both of parts and , and, if you wish, part
as well. Show all your
work. You may use any sources you
Mathematics 2260H Geometry I: Euclidean geometry
Trent University, Winter 2012
Quiz Solutions
Quiz #1. Tuesday, 17 January, 2012. [10 minutes]
1. Given a line segment AB, use (some of) Postulates IV, A, and S to show there exists
a line segment that is ex
Mathematics 226H Geometry I: Euclidean geometry
Trent University, Winter 2008
Solutions to Problem Set #9
1. (Exercise 3B.1) Line segments of length x and y have been drawn from vertex A of
ABC to base BC, and these make equal angles with the sides, as in
Mathematics 226H Geometry I: Euclidean geometry
Trent University, Winter 2008
Problem Set #11
Due on Friday, 4 April, 2008.
1. (Exercise 4B.2) Show using similar triangles that if Cevians AP , BQ, and CR are
parallel, then the Cevian product is trivial. [
Mathematics 226H Geometry I: Euclidean geometry
Trent University, Winter 2008
Solutions to Quizzes
Quiz #1. Friday, 18 January, 2008. [10 minutes]
1. Given a line segment AB, show, using Euclids system, that there is a point C so that
B is on AC and |BC|
Mathematics 2260H Geometry I: Euclidean geometry
Trent University, Winter 2011
Quiz Solutions
Quiz #1. Wednesday, 19 Thursday, 20 January, 2011 [10 minutes]
1. Given a line segment AB, construct a point C so that B is on AC and the length of
AC is twice t
Mathematics 226H Geometry I: Euclidean geometry
Trent University, Winter 2008
Solutions to Problem Set #7
1. (Exercise 2E.3) Show that in a right triangle, the inradiius, circumradius, and
semiperimeter are related by the formula s = r + 2R. [5]
Solution.
Mathematics 226H Geometry I: Euclidean geometry
Trent University, Winter 2008
Solutions to Problem Set #3
1. (Exercise 1E.3) Since a triangle is determined by angle-side-angle, there should be a
formula for KABC expressed in terms of a and B and C. Derive
Mathematics 226H Geometry I: Euclidean geometry
Trent University, Winter 2008
Solutions to Problem Set #5
1. (Exercise 2A.1) Show that quadrilateral ABCD can be inscribed in a circle if and
only if B and D are supplementary. [5]
Hint: To prove if, show th
Mathematics 226H Geometry I: Euclidean geometry
Trent University, Winter 2008
Take-Home Final Examination
Due on Friday, 25 April, 2008.
Instructions: Do all three of parts A C, and, if you wish, part
as well. Show all
your work. You may use your textbook
Mathematics 2260H Geometry I: Euclidean geometry
Trent University, Winter 2011
Take-Home Final Examination
Due on Tuesday, 26 April, 2011.
Instructions: Do both of parts A and B, and, if you wish, part as well. Show all your
work. You may use any sources
v
Mathematics 226H Geometry I: Euclidean geometry
Trent University, Winter 2008
Solutions to Problem Set #4
1. (Exercise 1H.5) Given two points X and Y on a circle, a point P is chosen on line
XY , outside of the circle, and tangent P T is drawn, where T
Mathematics 2260H Geometry I: Euclidean geometry
Trent University, Winter 2012
Solution to Assignment #3
Angle-Angle-Side!
1. Show that the Angle-Angle-Side congruence criterion actually works. That is, show
that if ABC = DEF , BCA = EF D, and CA = F D, t
Mathematics 2260H Geometry I: Euclidean geometry
Trent University, Winter 2012
Solution to Assignment #2
Angle-Side-Side?
1. Determine whether the Angle-Side-Side congruence criterion actually works. That is,
if ABC = DEF , BC = EF , and CA = F D, must it