The EPGY Theorem Proving Environment
Student Exercises
Geometry (M015)
The geometry course assigns approximately 20 proof exercises. In
working the exercises, a student has access to a database of Axioms,
Theorems, and Definitions. That database grows as the student
progresses in the course. Here is our current list.
Exercises
 Prove that if P, Q, and R are collinear then
R, Q, and P are collinear.
 Prove that if the point P lies on the line l then there is a point distinct from P that lies on l.
 Prove that given any pair of lines l and m there is a line n that intersects both l and m.
 Prove that there are are three distinct lines.
 Prove that if P and Q are distinct, A, B, P are collinear, A, B, Q are collinear, and P, Q, R are collinear then A, B, R are collinear.
 Prove that for all lines l, l l.
 Prove that if l m then m l.
 Prove that there are distinct lines l and m such that l m.
 Prove that if P is distinct from both Q and R, then
 if and only if
= .
 Prove that if P, Q, and R are not collinear, then
is not parallel to
.
 Prove: If
and P and R are on l, then Q is on l.
 Prove: If Q = R then
.
 Prove: There is a point P, not equal to Q, such that
.
 Prove: If
and
then
.
 Prove:
&
P = Q.
 Prove: If A B and P is on
then P is on
.
 Prove: If A B and P is on
then P is on
.
 Prove: If
and P is on
then
P is on .
 Prove: If
, P is on
, and
P is on then P = B.
 Prove: If
where A, B, and C are distinct and
P is on then
P is on or
P is on .
 Prove:
A, B, P are collinear.
Axioms
Axiom I1
Every line contains at least two distinct points.
Axiom I2
Every pair of points determines a line.
Axiom I3
There is at most one line containing any pair of distinct points.
Axiom D1
There are at least three points not all on the same line.
Axiom E
Two sets of points are equal if they have the same members.
Axiom P1
For any point P and line l there is a line through P parallel to
l.
Axiom P2
For any point P and line l there is at most one line through P parallel to l.
Axiom B1
If
, then the points P, Q, and R are collinear.
Axiom B4
If three points are collinear then one of them is between the other two.
Axiom B5
There is a point R, not equal to Q, such that
.
Axiom B8
If
C is not on l, A and C are on the same side of line l, and B and C are on the same side of line l, then A and B are on the same side of l.
Axiom C2 (Reflexivity)
Every segment is congruent to itself.
Axiom C3 (Symmetry)
If one segment is congruent to a second segment then the second segment is congruent to the first.
Axiom C4 (Transitivity)
If one segment is congruent to a second, and the second is congruent to a third, then the first segment is congruent to the third.
Axiom C6
Given distinct points P and Q and a segment
there is a point R on
such that
.
Axiom C7
If P and Q are distinct,
R, S are on , and
then R = S.
Axiom C9 (Reflexivity)
If A B and C B, then
ABC ABC.
Axiom C12
If A B, C B, and P Q, then there is a point R such that R Q and
PQR ABC.
Axiom C13
If
PQR is congruent to
PQS and the points R and S lie on the same side of
, then
= .
Axiom C14
If A, B and C are three noncollinear points, P, Q, and R are also noncollinear,
is congruent to
,
ABC is congruent to
PQR, and
is congruent to
, then
is congruent to
,
BAC is congruent to
QPR, and
ACB is congruent to
PRQ. (That is, every segment and angle of
ABC is congruent to the corresponding angle or segment of
PQR; the two triangles are congruent.)
Definitions
Definition 3.1
When P and Q are distinct points we write
to denote
the unique line containing P and Q.
Definition 3.2
We use the notation
P, Q, R are collinear to indicate that the points P,
Q, and R lie on the same line.
Definition 3.3
We will use
x and y intersect to indicate that there is a point P that
lies on both x and y.
Definition 3.4
The lines l and m are parallel, or l is parallel to m, written l m, if either l = m or l and m have no points in common.
Definition 3.5
Relations that are reflexive, symmetric, and transitive are called equivalence relations.
Definition 3.6
The line l is between points P and Q, written
, if for some A that lies on l,
.
Definition 3.7
Points P and Q are on the same side of l if
P is on l,
Q is on l, or
¬ .
Definition 3.8
For any pair of points P and Q, we let
denote the line segment joining P and Q; in particular, the points on
are exactly the points A such that
.
Note that, unlike
,
is defined in the case P = Q.
Definition 3.9
For distinct points P and Q, we let
denote the ray from P to Q; in particular, the points on
are exactly the points A such that A is between P and Q or Q is between P and A.
Definition 3.10
Ray
is between rays
and
if
and
intersect; this is denoted
. More formally,
Definition 3.11
Three points A, B, and C determine an angle, denoted
ABC, if B is different from both A and C. A point P is on
ABC, written
P is on ABC if P is on one of the rays
or
. More formally,
Definition 3.12
A point P is in the interior of
ABC, if P
is not on
ABC, P and C are on the same side of
, and P and A are on the same side of
.
Definition 3.13
Every three noncollinear points A, B, and C determine a triangle denoted
ABC. A point P is on
ABC if
P is on one of the segments
,
, or
.
Definition 3.14
A point P is in the interior of
ABC, written
P is in ,
if P and C are on the same side of
, P and A are
on the same side of
, and P and B are on the same side
of
.
Theorems
Theorem 3.1
If P and Q are distinct then they both lie on
.
Theorem 3.2
If P and Q are distinct and A and B are distinct points on the line
then
= .
Theorem 3.3
If P, Q, and R are not distinct (that is, at least two of these
points are equal) then they are collinear.
Theorem 3.4
If P, Q, and R are collinear then P, R, and Q are collinear and Q, P, and R are collinear and so on.
Theorem 3.5
Suppose A B. Then A, B, and C are collinear if and only if C is on
.
Theorem 3.6
If A B, A, B, P are collinear, A, B, Q are collinear, and A, B, R are collinear then P, Q, R are collinear.
Theorem 3.7
If P and Q are distinct, A, B, P are collinear, A, B, Q are collinear, and P, Q, R are collinear then A, B, R are collinear.
Theorem 3.9
Either l m or l and m have exactly one point in common.
Theorem 3.10
For all lines l, l l; we say that the relation  is reflexive
Theorem 3.11
If l m then m l; we say that the relation  is symmetric
Theorem 3.12
If l m and m n then l n. We say that the relation  is transitive
Corollary 3.13
The relation  is an equivalence relation.
Theorem 3.14
If l m, l n and n intersects l then n intersects m.
Theorem 3.15
Distinct points P, Q, and R are collinear if and only if
 .
Lemma 3.16
If
and P and R are on l, then Q is on l.
Theorem 3.21
Let A, B, and C be three noncollinear points, and l a line
that does not pass through any of them. Suppose further that l is
between A and B. Then l is either between A and C, or
between B and C, but not both.
Theorem 3.22
For any three points A, B, and C, two points are on the same side of the line l.
Theorem 3.24
If P and R are distinct then there is a point Q, not equal to P or R, such that Q is between P and R.
Lemma 3.25
For any points A and B, both A and B lie on
.
Lemma 3.30
Suppose A B. Then there exist points P and Q, distinct from B, that lie on
, and such that
. On the other hand, there do not exist such points for A.
Theorem 3.31
Segments
and
are equal if and only if both A = P and B = Q, or both A = Q and B = P.
Theorem 3.32
Suppose A B and B C. Then rays
and
are equal if and only if A = C.
Theorem 3.33
Suppose A B and A C. Then rays
and
are equal if and only if C is on
.
Theorem 3.34
Suppose P Q. Then rays
and
are equal if and only if A = P and Q is on
.
Theorem 3.37
If P is distinct from each of the points A, B, and C, and
= then
.
Theorem 3.38
If P is not equal to A and not equal to B then
.
Theorem 3.39
If
and A, B and P are not collinear then
. More formally,
Theorem 3.40
If A, B, P are not collinear, P C,
, and A, B, C are collinear, then
. More formally,
Lemma 3.46
Suppose that
,
,
, and B and D are on the same side of
. Then
and
.
Theorem 3.47
If B is different from both A and C then
ABC and
CBA denote the same angle.
Theorem 3.48
The statement ``
P is in the interior of
ABC'' is equivalent to the statement `` P does not lie on
ABC and
is between
and
.''
Theorem 3.49
If A, B, and C are not collinear, then
ABC,
ACB,
BAC,
BCA,
CAB, and
CBA all denote the same thing.
Theorem 3.50
If A, B, and C are not collinear and the points D, E, and F are not collinear then
ABC = DEF if and only if the points A, B, and C and the points D, E, and F are the same sets of three points. This is expressed formally by,
Theorem 3.51
If A, B, and C are noncollinear then saying that
P is in
is equivalent to saying that P is on the interior of each of the
angles
ABC,
BCA, and
CAB.
Theorem 3.52
If A, B, and C are noncollinear then saying that
P is in
is equivalent to saying that P is on the interior of the angles
ABC and
BCA
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Last modified: Mon Dec 10 09:38:02 PST 2001
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