Math 460: Homework # 9. Due Thurs, April 17
1. Finish the proof of theorem 43. (One direction is left.)
2. (See gure below.) Given: H is the orthocenter of ABC, and the points U , V ,
W , and P are the midpoints of the segments AH, CH, BH, and AB. Prove t
DEFINITIONS
Degree A degree is the
1
th
180
part of a straight angle.
Right Angle A 90 angle is called a right angle.
Perpendicular Two lines are called perpendicular if they form a right angle.
Congruent Triangles Two triangles ABC and DEF are congruent
Math 460: Homework # 10. Due November 1
1. (Use Geometers Sketchpad.) Begin with a point A and four lines , m, n and p
that go through A. Next, hide the points other than A used to construct these
lines (this is important). Choose a point B on and a point
Math 460: Homework # 9. Due Friday October 18
1. (Use Geometers Sketchpad.) For most quadrilaterals, the four angle bisectors are
not concurrent.
(a) Find an equation that the sides of the quadrilateral have to satisfy if the four
angle bisectors are conc
Math 460: Homework # 8. Due Friday October 11
1. (Use Geometers Sketchpad.)
(a) Make a script which constructs the orthocenter of a given triangle. Print out a
copy.
(b) Construct a triangle ABC and play your script to nd the orthocenter. Label
the orthoc
Math 460: Homework # 7. Due Friday October 4
1. (Use Geometers Sketchpad.) Start with triangle ABC . Draw a line that crosses
all three lines AB , AC and BC , and let P , Q and R be the intersections of with
the lines AB , AC and BC respectively. Next dra
Math 460: Homework # 6. Due Friday September 27
1. (Use Geometers Sketchpad.) Consider the following algorithm for constructing a
triangle with three given sides, using Circle by center and radius and Segment
commands: Start with 3 line segments XY , ZW a
Math 460: Homework # 5. Due Friday, September 20
1. (Use Geometers Sketchpad.) Draw a triangle ABC . Then draw a line through A
parallel to BC , a line m through B parallel to AC , and a line n through C parallel
to AB . Let D be the intersection of and m
Math 460: Homework # 10. Due Thurs. April 24
1. See Figure 1. Prove that
ABBCC A
= 1.
AC BAC B
B'
C
P
A'
A
B
C'
Figure 1
2. See Figure 2. Given: P C is tangent to the circle. To prove: P A P B = P C 2 .
(This is a special case of theorem 42.)
B
P
A
C
Figu
Math 460: Homework # 8. Due Thurs April 3
1. (See Figure 1.) Given: M , N and P are the midpoints of AB, AC and BC. To
AD
1
prove:
= .
DB
2
C
N
P
F
E
A
D
M
B
Figure 1
2. (See Figure 2.) Given: the things that look like squares are squares. To prove: the
a
Math 460: Homework # 5. Due Thurs. Feb 20
1. (See Figure 1.) Prove the case of theorem E (from analytic geometry notes) where
A lies in the second quadrant, and B in the rst.
C=A+B
A
B
O
Figure 1:
2. Prove that if ABC and DEF are two triangles with
AC
BC
Course Notes for MA 460. Version 5.
Jim McClure
1
Denitions and Basic Facts.
The goal of this course is to help you become expert in geometry, so that you can teach
it with condence and pleasure. We begin more or less where you left o in high school,
and
Math 460: Homework # 1. Due Thurs. Jan 23
Rules for writing up proofs on the homework:
Any fact you use must be from the Course Notes or from previous homework (but
not from the Geometers Sketchpad problems).
You must give a justication for every step i
MA 460 Supplement: Analytic geometry
Donu Arapura
In the 1600s Descartes introduced cartesian coordinates which changed the way we
now do geometry. This also paved for subsequent developments such as calculus. Here
we revisit some parts of Euclidean geome
MA 460 Supplement: spherical geometry
Donu Arapura
Although spherical geometry is not as old or as well known as Euclidean geometry, it
is quite old and quite beautiful. The original motivation probably came from astronomy
and navigation, where stars in t
MA 460 Supplement: Cevas theorem using analytic
geometry
Donu Arapura
Cevas theorem, which is theorem 36 in McClure, says
Theorem 36. Suppose that ABC is a triangle, and let A , B and C be points on the
lines BC, AC and AB other than the vertices. If AA ,
Math 460: Homework # 6. Due Thurs. March 13
1. (See Figure 1.) Given: M , N and P are the midpoints of AB, AC and BC
respectively, M D is parallel to AP , and M D = AP . To prove: CD = N B. (Hint:
there are three parallelograms in this picture. Do not dra
Math 460: Homework # 2. Due Thurs. Jan 30
1. (See Figure 1) Given that AB is perpendicular to AC, that AD is perpendicular
to BC, and that AB = BE, prove that 1 = 2.
C
E
D
2
1
B
A
Figure 1
2. Given a quadrilateral ABCD with AB = BC and CD = AD, prove that
Math 460: Homework # 3. Due Thurs. Feb 6
You are allowed to use any theorem in the notes.
1. (See Figure 1.) For this problem you need the denition of circle: a circle consists
of all of the points which are at a given distance (called the radius) from a
Math 460: Homework # 4. Due Thurs. Feb 13
1. Use Geometers Sketchpad to construct a triangle, along with the following
(a) its circumcenter (labeled O)
(b) its incenter (labeled I)
(c) its orthocenter (labeled H)
(d) its centroid (labeled G)
(e) the line
Math 460: Homework # 7. Due Thurs. March 27
1. Prove case (ii) of Menelaus theorem (theorem 31).
2. This problem, which uses analytic geometry, is to show that obvious converse to
Menelaus is false. Let A = (0, 0), B = (2, 0), C = (0, 2). Find A on the li
Math 460: Homework # 4. Due Friday September 13
Note: Problems 1, 2, and 5 all use Geometers Sketchpad.
1. (Use Geometers Sketchpad.) Construct a quadrilateral ABCD and let M, N, P
and Q be the midpoints of the sides. Find the areas of the quadrilaterals
Math 460: Homework # 5. Due Friday, September 20
1. (Use Geometers Sketchpad.) Draw a triangle ABC . Then draw a line through A
parallel to BC , a line m through B parallel to AC , and a line n through C parallel
to AB . Let D be the intersection of and m