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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 3.43  

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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