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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 $\;\stackrel{{\longleftrightarrow}}{{PQ}}\;$ || $\;\stackrel{{\longleftrightarrow}}{{PR}}\;$ if and only if $\;\stackrel{{\longleftrightarrow}}{{PQ}}\;$ = $\;\stackrel{{\longleftrightarrow}}{{PR}}\;$ .
Prove that if P, Q, and R are not collinear, then $\;\stackrel{{\longleftrightarrow}}{{PQ}}\;$ is not parallel to $\;\stackrel{{\longleftrightarrow}}{{PR}}\;$ .
Prove: If $\mbox{$P$-$Q$-$R$}$ and P and R are on l, then Q is on l.
Prove: If Q = R then $\mbox{$P$-$Q$-$R$}$ .
Prove: There is a point P, not equal to Q, such that $\mbox{$P$-$Q$-$R$}$ .
Prove: If $\mbox{$P$-$Q$-$S$}$ and $\mbox{$Q$-$R$-$S$}$ then $\mbox{$P$-$R$-$S$}$ .
Prove: $\mbox{$P$-$Q$-$R$}$ & $\mbox{$Q$-$P$-$R$}$ $\rightarrow$ P = Q.
Prove: If A $\neq$ B and P is on $\;\stackrel{{\mbox{\rule{4mm}{.2mm}}}}{{AB}}\;$ then P is on $\;\stackrel{{\longrightarrow}}{{AB}}\;$ .
Prove: If A $\neq$ B and P is on $\;\stackrel{{\longrightarrow}}{{AB}}\;$ then P is on $\;\stackrel{{\longleftrightarrow}}{{AB}}\;$ .
Prove: If $\mbox{$A$-$B$-$C$}$ and P is on $\;\stackrel{{\mbox{\rule{4mm}{.2mm}}}}{{AB}}\;$ then P is on $\;\stackrel{{\mbox{\rule{4mm}{.2mm}}}}{{AC}}\;$ .
Prove: If $\mbox{$A$-$B$-$C$}$ , P is on $\;\stackrel{{\mbox{\rule{4mm}{.2mm}}}}{{AB}}\;$ , and P is on $\;\stackrel{{\mbox{\rule{4mm}{.2mm}}}}{{BC}}\;$ then P = B.
Prove: If $\mbox{$A$-$B$-$C$}$ where A, B, and C are distinct and P is on $\;\stackrel{{\longleftrightarrow}}{{AC}}\;$ then P is on $\;\stackrel{{\longrightarrow}}{{BA}}\;$ or P is on $\;\stackrel{{\longrightarrow}}{{BC}}\;$ .
Prove: $\mbox{$\mbox{}\stackrel{\longrightarrow}{PA}\mbox{}$-$\mbox{}\stackrel{\longrightarrow}{PB}\mbox{}$-$\mbox{}\stackrel{\longrightarrow}{PA}\mbox{}$}$ $\rightarrow$ 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 $\mbox{$P$-$Q$-$R$}$ , then the points P, Q, and R are collinear.

Axiom B2

$% latex2html id marker 6517 $\displaystyle \mbox{$P$-$Q$-$R$}$$ $% latex2html id marker 6518 $\displaystyle \mbox{ $\rightarrow$\ }$$ $% latex2html id marker 6519 $\displaystyle \mbox{$R$-$Q$-$P$}$$ .

Axiom B3

$% latex2html id marker 6522 $\displaystyle \mbox{$P$-$Q$-$P$}$$ $% latex2html id marker 6523 $\displaystyle \mbox{ $\rightarrow$\ }$$ 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 $\mbox{$P$-$Q$-$R$}$ .

Axiom B6 If $\mbox{$P$-$Q$-$S$}$ and $\mbox{$Q$-$R$-$S$}$ , then $\mbox{$P$-$Q$-$R$}$ .

Axiom B7 If Q $\neq$ R, $\mbox{$P$-$Q$-$R$}$ and $\mbox{$Q$-$R$-$S$}$ , then $\mbox{$P$-$Q$-$S$}$ .

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 $\;\stackrel{{\mbox{\rule{4mm}{.2mm}}}}{{AA}}\;$ is congruent to $\;\stackrel{{\mbox{\rule{4mm}{.2mm}}}}{{BC}}\;$ 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 $\mbox{$A$-$B$-$C$}$ , $\mbox{$P$-$Q$-$R$}$ , $\;\stackrel{{\mbox{\rule{4mm}{.2mm}}}}{{AB}}\;$ $\cong$ $\;\stackrel{{\mbox{\rule{4mm}{.2mm}}}}{{PQ}}\;$ , and $\;\stackrel{{\mbox{\rule{4mm}{.2mm}}}}{{BC}}\;$ $\cong$ $\;\stackrel{{\mbox{\rule{4mm}{.2mm}}}}{{QR}}\;$ then $\;\stackrel{{\mbox{\rule{4mm}{.2mm}}}}{{AC}}\;$ $\cong$ $\;\stackrel{{\mbox{\rule{4mm}{.2mm}}}}{{PR}}\;$ .

Axiom C6 Given distinct points P and Q and a segment $\;\stackrel{{\mbox{\rule{4mm}{.2mm}}}}{{AB}}\;$ there is a point R on $\;\stackrel{{\longrightarrow}}{{PQ}}\;$ such that $\;\stackrel{{\mbox{\rule{4mm}{.2mm}}}}{{PR}}\;$ $\cong$ $\;\stackrel{{\mbox{\rule{4mm}{.2mm}}}}{{AB}}\;$ .

Axiom C7 If P and Q are distinct, R, S are on $\;\stackrel{{\longrightarrow}}{{PQ}}\;$ , and $\;\stackrel{{\mbox{\rule{4mm}{.2mm}}}}{{PR}}\;$ $\cong$ $\;\stackrel{{\mbox{\rule{4mm}{.2mm}}}}{{PS}}\;$ then R = S.

Axiom C8 If $\angle$ ABA $\cong$ $\angle$ PQR then $\;\stackrel{{\longrightarrow}}{{QP}}\;$ = $\;\stackrel{{\longrightarrow}}{{QR}}\;$ .

Axiom C9 (Reflexivity) If A $\neq$ B and C $\neq$ B, then $\angle$ ABC $\cong$ $\angle$ ABC.

Axiom C10 (Symmetry) If $\angle$ ABC $\cong$ $\angle$ PQR then $\angle$ PQR $\cong$ $\angle$ ABC.

Axiom C11 (Transitivity) If $\angle$ ABC $\cong$ $\angle$ PQR and $\angle$ PQR $\cong$ $\angle$ DEF, then $\angle$ ABC $\cong$ $\angle$ DEF.

Axiom C12 If A $\neq$ B, C $\neq$ B, and P $\neq$ Q, then there is a point R such that R $\neq$ Q and $\angle$ PQR $\cong$ $\angle$ ABC.

Axiom C13 If $\angle$ PQR is congruent to $\angle$ PQS and the points R and S lie on the same side of $\;\stackrel{{\longleftrightarrow}}{{PQ}}\;$ , then $\;\stackrel{{\longrightarrow}}{{QS}}\;$ = $\;\stackrel{{\longrightarrow}}{{QR}}\;$ .

Axiom C14 If A, B and C are three noncollinear points, P, Q, and R are also noncollinear, $\;\stackrel{{\mbox{\rule{4mm}{.2mm}}}}{{AB}}\;$ is congruent to $\;\stackrel{{\mbox{\rule{4mm}{.2mm}}}}{{PQ}}\;$ , $\angle$ ABC is congruent to $\angle$ PQR, and $\;\stackrel{{\mbox{\rule{4mm}{.2mm}}}}{{BC}}\;$ is congruent to $\;\stackrel{{\mbox{\rule{4mm}{.2mm}}}}{{QR}}\;$ , then $\;\stackrel{{\mbox{\rule{4mm}{.2mm}}}}{{AC}}\;$ is congruent to $\;\stackrel{{\mbox{\rule{4mm}{.2mm}}}}{{PR}}\;$ , $\angle$ BAC is congruent to $\angle$ QPR, and $\angle$ ACB is congruent to $\angle$ PRQ. (That is, every segment and angle of $\triangle$ ABC is congruent to the corresponding angle or segment of $\triangle$ PQR; the two triangles are congruent.)

Definitions

Definition 3.1 When P and Q are distinct points we write $\;\stackrel{{\longleftrightarrow}}{{PQ}}\;$ 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 $\mbox{$P$-$l$-$Q$}$ , if for some A that lies on l, $\mbox{$P$-$A$-$Q$}$ .

Definition 3.7 Points P and Q are on the same side of l if P is on l, Q is on l, or ¬ $\mbox{$P$-$l$-$Q$}$ .

Definition 3.8 For any pair of points P and Q, we let $\;\stackrel{{\mbox{\rule{4mm}{.2mm}}}}{{PQ}}\;$ denote the line segment joining P and Q; in particular, the points on $\;\stackrel{{\mbox{\rule{4mm}{.2mm}}}}{{PQ}}\;$ are exactly the points A such that $\mbox{$P$-$A$-$Q$}$ . Note that, unlike $\;\stackrel{{\longleftrightarrow}}{{PQ}}\;$ , $\;\stackrel{{\mbox{\rule{4mm}{.2mm}}}}{{PQ}}\;$ is defined in the case P = Q.

Definition 3.9 For distinct points P and Q, we let $\;\stackrel{{\longrightarrow}}{{PQ}}\;$ denote the ray from P to Q; in particular, the points on $\;\stackrel{{\longrightarrow}}{{PQ}}\;$ are exactly the points A such that A is between P and Q or Q is between P and A.

Definition 3.10 Ray $\;\stackrel{{\longrightarrow}}{{PB}}\;$ is between rays $\;\stackrel{{\longrightarrow}}{{PA}}\;$ and $\;\stackrel{{\longrightarrow}}{{PC}}\;$ if $\;\stackrel{{\mbox{\rule{4mm}{.2mm}}}}{{AC}}\;$ and $\;\stackrel{{\longrightarrow}}{{PB}}\;$ intersect; this is denoted $\mbox{$\mbox{}\stackrel{\longrightarrow}{PA}\mbox{}$-$\mbox{}\stackrel{\longrightarrow}{PB}\mbox{}$-$\mbox{}\stackrel{\longrightarrow}{PC}\mbox{}$}$ . More formally,

Definition 3.11 Three points A, B, and C determine an angle, denoted $\angle$ ABC, if B is different from both A and C. A point P is on $\angle$ ABC, written P is on $\angle$ ABC if P is on one of the rays $\;\stackrel{{\longrightarrow}}{{BA}}\;$ or $\;\stackrel{{\longrightarrow}}{{BC}}\;$ . More formally,

Definition 3.12 A point P is in the interior of $\angle$ ABC, if P is not on $\angle$ ABC, P and C are on the same side of $\;\stackrel{{\longleftrightarrow}}{{BA}}\;$ , and P and A are on the same side of $\;\stackrel{{\longleftrightarrow}}{{BC}}\;$ .

Definition 3.13 Every three non-collinear points A, B, and C determine a triangle denoted $\triangle$ ABC. A point P is on $\triangle$ ABC if P is on one of the segments $\;\stackrel{{\mbox{\rule{4mm}{.2mm}}}}{{AB}}\;$ , $\;\stackrel{{\mbox{\rule{4mm}{.2mm}}}}{{BC}}\;$ , or $\;\stackrel{{\mbox{\rule{4mm}{.2mm}}}}{{CA}}\;$ .

Definition 3.14 A point P is in the interior of $\triangle$ ABC, written

P is in $% latex2html id marker 6872 $\displaystyle \mbox{\rm\ the interior of $\triangle ABC$}$$ ,

if P and C are on the same side of $\;\stackrel{{\longleftrightarrow}}{{AB}}\;$ , P and A are on the same side of $\;\stackrel{{\longleftrightarrow}}{{BC}}\;$ , and P and B are on the same side of $\;\stackrel{{\longleftrightarrow}}{{CA}}\;$ .

Theorems

Theorem 3.1 If P and Q are distinct then they both lie on $\;\stackrel{{\longleftrightarrow}}{{PQ}}\;$ .

Theorem 3.2 If P and Q are distinct and A and B are distinct points on the line $\;\stackrel{{\longleftrightarrow}}{{PQ}}\;$ then $\;\stackrel{{\longleftrightarrow}}{{AB}}\;$ = $\;\stackrel{{\longleftrightarrow}}{{PQ}}\;$ .

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 $\neq$ B. Then A, B, and C are collinear if and only if C is on $\;\stackrel{{\longleftrightarrow}}{{AB}}\;$ .

Theorem 3.6 If A $\neq$ 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 $\neq$ n and n intersects l then n intersects m.

Theorem 3.15 Distinct points P, Q, and R are collinear if and only if $\;\stackrel{{\longleftrightarrow}}{{PQ}}\;$ || $\;\stackrel{{\longleftrightarrow}}{{QR}}\;$ .

Lemma 3.16 If $\mbox{$P$-$Q$-$R$}$ and P and R are on l, then Q is on l.

Lemma 3.17

P = Q $% latex2html id marker 6983 $\displaystyle \mbox{ $\rightarrow$\ }$$ $% latex2html id marker 6984 $\displaystyle \mbox{$P$-$Q$-$R$}$$ .

Lemma 3.18

$% latex2html id marker 6987 $\displaystyle \mbox{$P$-$R$-$S$}$$ & $% latex2html id marker 6988 $\displaystyle \mbox{$P$-$Q$-$R$}$$ $% latex2html id marker 6989 $\displaystyle \mbox{ $\rightarrow$\ }$$ $% latex2html id marker 6990 $\displaystyle \mbox{$P$-$Q$-$S$}$$ & $% latex2html id marker 6991 $\displaystyle \mbox{$Q$-$R$-$S$}$$ .

Lemma 3.19

$% latex2html id marker 6994 $\displaystyle \mbox{$P$-$Q$-$R$}$$ & $% latex2html id marker 6995 $\displaystyle \mbox{$Q$-$P$-$R$}$$ $% latex2html id marker 6996 $\displaystyle \mbox{ $\rightarrow$\ }$$ P = Q.

Theorem 3.20

Q $\displaystyle \neq$ R & $% latex2html id marker 7000 $\displaystyle \mbox{$P$-$Q$-$R$}$$ & $% latex2html id marker 7001 $\displaystyle \mbox{$Q$-$R$-$S$}$$ $% latex2html id marker 7002 $\displaystyle \mbox{ $\rightarrow$\ }$$ $% latex2html id marker 7003 $\displaystyle \mbox{$P$-$R$-$S$}$$ .

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 $\displaystyle \neq$ B & $% latex2html id marker 7025 $\displaystyle \mbox{$A$-$B$-$P$}$$ & $% latex2html id marker 7026 $\displaystyle \mbox{$A$-$B$-$Q$}$$ $% latex2html id marker 7027 $\displaystyle \mbox{ $\rightarrow$\ }$$ ( $% latex2html id marker 7028 $\displaystyle \mbox{$B$-$P$-$Q$}$$ $% latex2html id marker 7029 $\displaystyle \mbox{ $\vee$\ }$$ $% latex2html id marker 7030 $\displaystyle \mbox{$B$-$Q$-$P$}$$ ).

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 $\;\stackrel{{\mbox{\rule{4mm}{.2mm}}}}{{AB}}\;$ .

Lemma 3.26 For any points A and B, $\;\stackrel{{\mbox{\rule{4mm}{.2mm}}}}{{AB}}\;$ = $\;\stackrel{{\mbox{\rule{4mm}{.2mm}}}}{{BA}}\;$ .

Lemma 3.27 If A = B and P is on $\;\stackrel{{\mbox{\rule{4mm}{.2mm}}}}{{AB}}\;$ then A = P.

Lemma 3.28 If A $\neq$ B, then A, B are on $\;\stackrel{{\longrightarrow}}{{AB}}\;$ .

Lemma 3.29 If C is on $\;\stackrel{{\longrightarrow}}{{AB}}\;$ and A $\neq$ C, then B is on $\;\stackrel{{\longrightarrow}}{{AC}}\;$ .

Lemma 3.30 Suppose A $\neq$ B. Then there exist points P and Q, distinct from B, that lie on $\;\stackrel{{\longrightarrow}}{{AB}}\;$ , and such that $\mbox{$P$-$B$-$Q$}$ . On the other hand, there do not exist such points for A.

Theorem 3.31 Segments $\;\stackrel{{\mbox{\rule{4mm}{.2mm}}}}{{AB}}\;$ and $\;\stackrel{{\mbox{\rule{4mm}{.2mm}}}}{{PQ}}\;$ are equal if and only if both A = P and B = Q, or both A = Q and B = P.

Theorem 3.32 Suppose A $\neq$ B and B $\neq$ C. Then rays $\;\stackrel{{\longrightarrow}}{{AB}}\;$ and $\;\stackrel{{\longrightarrow}}{{CB}}\;$ are equal if and only if A = C.

Theorem 3.33 Suppose A $\neq$ B and A $\neq$ C. Then rays $\;\stackrel{{\longrightarrow}}{{AB}}\;$ and $\;\stackrel{{\longrightarrow}}{{AC}}\;$ are equal if and only if C is on $\;\stackrel{{\longrightarrow}}{{AB}}\;$ .

Theorem 3.34 Suppose P $\neq$ Q. Then rays $\;\stackrel{{\longrightarrow}}{{AB}}\;$ and $\;\stackrel{{\longrightarrow}}{{PQ}}\;$ are equal if and only if A = P and Q is on $\;\stackrel{{\longrightarrow}}{{AB}}\;$ .

Theorem 3.35 If A $\neq$ B and P is on $\;\stackrel{{\mbox{\rule{4mm}{.2mm}}}}{{AB}}\;$ then P is on $\;\stackrel{{\longrightarrow}}{{AB}}\;$ .

Theorem 3.36 If A $\neq$ B and P is on $\;\stackrel{{\longrightarrow}}{{AB}}\;$ then P is on $\;\stackrel{{\longleftrightarrow}}{{AB}}\;$ .

Theorem 3.37 If P is distinct from each of the points A, B, and C, and $\;\stackrel{{\longrightarrow}}{{PA}}\;$ = $\;\stackrel{{\longrightarrow}}{{PB}}\;$ then $\mbox{$\mbox{}\stackrel{\longrightarrow}{PA}\mbox{}$-$\mbox{}\stackrel{\longrightarrow}{PB}\mbox{}$-$\mbox{}\stackrel{\longrightarrow}{PC}\mbox{}$}$ .

Theorem 3.38 If P is not equal to A and not equal to B then $\mbox{$\mbox{}\stackrel{\longrightarrow}{PA}\mbox{}$-$\mbox{}\stackrel{\longrightarrow}{PA}\mbox{}$-$\mbox{}\stackrel{\longrightarrow}{PB}\mbox{}$}$ .

Theorem 3.39 If $\mbox{$A$-$B$-$C$}$ and A, B and P are not collinear then $\mbox{$\mbox{}\stackrel{\longrightarrow}{PA}\mbox{}$-$\mbox{}\stackrel{\longrightarrow}{PB}\mbox{}$-$\mbox{}\stackrel{\longrightarrow}{PC}\mbox{}$}$ . More formally,

Theorem 3.40 If A, B, P are not collinear, P $\neq$ C, $\mbox{$\mbox{}\stackrel{\longrightarrow}{PA}\mbox{}$-$\mbox{}\stackrel{\longrightarrow}{PB}\mbox{}$-$\mbox{}\stackrel{\longrightarrow}{PC}\mbox{}$}$ , and A, B, C are collinear, then $\mbox{$A$-$B$-$C$}$ . More formally,

Theorem 3.41 If $\mbox{$\mbox{}\stackrel{\longrightarrow}{PA}\mbox{}$-$\mbox{}\stackrel{\longrightarrow}{PB}\mbox{}$-$\mbox{}\stackrel{\longrightarrow}{PC}\mbox{}$}$ , then $\mbox{$\mbox{}\stackrel{\longrightarrow}{PC}\mbox{}$-$\mbox{}\stackrel{\longrightarrow}{PB}\mbox{}$-$\mbox{}\stackrel{\longrightarrow}{PA}\mbox{}$}$ .

Lemma 3.42

$% latex2html id marker 7186 $\displaystyle \mbox{$\mbox{}\stackrel{\longrightarr... ...el{\longrightarrow}{PB}\mbox{}$-$\mbox{}\stackrel{\longrightarrow}{PA}\mbox{}$}$$ $% latex2html id marker 7187 $\displaystyle \mbox{ $\rightarrow$\ }$$ $% latex2html id marker 7188 $\displaystyle \mbox{\rm$A$, $B$, $P$\ are collinear}$$ .

Lemma 3.43

P $\displaystyle \neq$ A & P $\displaystyle \neq$ B & P $\displaystyle \neq$ C & (A = B $% latex2html id marker 7194 $\displaystyle \mbox{ $\vee$\ }$$ B = C) $% latex2html id marker 7195 $\displaystyle \mbox{ $\rightarrow$\ }$$ $% latex2html id marker 7196 $\displaystyle \mbox{$\mbox{}\stackrel{\longrightarr... ...el{\longrightarrow}{PB}\mbox{}$-$\mbox{}\stackrel{\longrightarrow}{PC}\mbox{}$}$$ .

Lemma 3.44

¬( $% latex2html id marker 7199 $\displaystyle \mbox{\rm$P$, $A$, $C$\ are collinear}$$ ) $% latex2html id marker 7200 $\displaystyle \mbox{ $\rightarrow$\ }$$ $% latex2html id marker 7201 $\displaystyle \exists$$ B( $\displaystyle \;\stackrel{{\longrightarrow}}{{PA}}\;$ $\displaystyle \neq$ $\displaystyle \;\stackrel{{\longrightarrow}}{{PB}}\;$ & $\displaystyle \;\stackrel{{\longrightarrow}}{{PB}}\;$ $\displaystyle \neq$ $\displaystyle \;\stackrel{{\longrightarrow}}{{PC}}\;$ & $% latex2html id marker 7208 $\displaystyle \mbox{$\mbox{}\stackrel{\longrightarr... ...el{\longrightarrow}{PB}\mbox{}$-$\mbox{}\stackrel{\longrightarrow}{PC}\mbox{}$}$$ ).

Lemma 3.45

$% latex2html id marker 7211 $\displaystyle \mbox{$\mbox{}\stackrel{\longrightarr... ...el{\longrightarrow}{PB}\mbox{}$-$\mbox{}\stackrel{\longrightarrow}{PD}\mbox{}$}$$ & $% latex2html id marker 7212 $\displaystyle \mbox{$\mbox{}\stackrel{\longrightarr... ...el{\longrightarrow}{PC}\mbox{}$-$\mbox{}\stackrel{\longrightarrow}{PD}\mbox{}$}$$ $% latex2html id marker 7213 $\displaystyle \mbox{ $\rightarrow$\ }$$ $% latex2html id marker 7214 $\displaystyle \mbox{$\mbox{}\stackrel{\longrightarr... ...el{\longrightarrow}{PB}\mbox{}$-$\mbox{}\stackrel{\longrightarrow}{PC}\mbox{}$}$$ & $% latex2html id marker 7215 $\displaystyle \mbox{$\mbox{}\stackrel{\longrightarr... ...el{\longrightarrow}{PC}\mbox{}$-$\mbox{}\stackrel{\longrightarrow}{PD}\mbox{}$}$$ .

Lemma 3.46 Suppose that $\mbox{$\mbox{}\stackrel{\longrightarrow}{PA}\mbox{}$-$\mbox{}\stackrel{\longrightarrow}{PB}\mbox{}$-$\mbox{}\stackrel{\longrightarrow}{PC}\mbox{}$}$ , $\mbox{$\mbox{}\stackrel{\longrightarrow}{PB}\mbox{}$-$\mbox{}\stackrel{\longrightarrow}{PC}\mbox{}$-$\mbox{}\stackrel{\longrightarrow}{PD}\mbox{}$}$ , $\;\stackrel{{\longrightarrow}}{{PB}}\;$ $\neq$ $\;\stackrel{{\longrightarrow}}{{PC}}\;$ , and B and D are on the same side of $\;\stackrel{{\longleftrightarrow}}{{PA}}\;$ . Then $\mbox{$\mbox{}\stackrel{\longrightarrow}{PA}\mbox{}$-$\mbox{}\stackrel{\longrightarrow}{PB}\mbox{}$-$\mbox{}\stackrel{\longrightarrow}{PD}\mbox{}$}$ and $\mbox{$\mbox{}\stackrel{\longrightarrow}{PA}\mbox{}$-$\mbox{}\stackrel{\longrightarrow}{PC}\mbox{}$-$\mbox{}\stackrel{\longrightarrow}{PD}\mbox{}$}$ .

Theorem 3.47 If B is different from both A and C then $\angle$ ABC and $\angle$ CBA denote the same angle.

Theorem 3.48 The statement `` P is in the interior of $\angle$ ABC'' is equivalent to the statement ``P does not lie on $\angle$ ABC and $\;\stackrel{{\longrightarrow}}{{BP}}\;$ is between $\;\stackrel{{\longrightarrow}}{{BA}}\;$ and $\;\stackrel{{\longrightarrow}}{{BC}}\;$ .''

Theorem 3.49 If A, B, and C are not collinear, then $\triangle$ ABC, $\triangle$ ACB, $\triangle$ BAC, $\triangle$ BCA, $\triangle$ CAB, and $\triangle$ 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 $\triangle$ ABC = $\triangle$ 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 $% latex2html id marker 7293 $\displaystyle \mbox{\rm\ the interior of $\triangle ABC$}$$

is equivalent to saying that P is on the interior of each of the angles $\angle$ ABC, $\angle$ BCA, and $\angle$ CAB.

Theorem 3.52 If A, B, and C are noncollinear then saying that

P is in $% latex2html id marker 7306 $\displaystyle \mbox{\rm\ the interior of $\triangle ABC$}$$

is equivalent to saying that P is on the interior of the angles $\angle$ ABC and $\angle$ BCA

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Copyright © 1993, 1994, 1995, 1996, Nikos Drakos, Computer Based Learning Unit, University of Leeds.
Copyright © 1997, 1998, 1999, Ross Moore, Mathematics Department, Macquarie University, Sydney.