# 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 B2

.

Axiom B3

P = Q.

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 B6   If and , then .

Axiom B7   If Q R, and , then .

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 C1   If     is congruent to     then B = C.

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 C5   If , ,        , and         then        .

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 C8   If ABA PQR then     =    .

Axiom C9 (Reflexivity)   If A B and C B, then ABC ABC.

Axiom C10 (Symmetry)   If ABC PQR then PQR ABC.

Axiom C11 (Transitivity)   If ABC PQR and PQR DEF, then ABC DEF.

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 non-collinear 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 3.8   There are (at least) two distinct lines.

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.

Lemma 3.17

P = Q.

Lemma 3.18

& & .

Lemma 3.19

& P = Q.

Theorem 3.20

Q R & & .

Theorem 3.21   Let A, B, and C be three non-collinear 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 3.23

A B & & ().

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 3.26   For any points A and B,     =    .

Lemma 3.27   If A = B and P is on     then A = P.

Lemma 3.28   If A B, then A, B are on    .

Lemma 3.29   If C is on     and A C, then B is 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 3.35   If A B and P is on     then P is on    .

Theorem 3.36   If A B and P is on     then P 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,

Theorem 3.41   If , then .

Lemma 3.43

P A & P B & P C & (A = BB = C).

Lemma 3.44

¬()B      &         & ).

Lemma 3.45

& & .

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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Copyright © 1997, 1998, 1999, Ross Moore, Mathematics Department, Macquarie University, Sydney.

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