



In
geometry, we need to prove stuff is true.



Most
people think this proof stuff is really weird and wacky. 


Here's
what's really going on. 





It
works like a detective story. 





Before
they get to be real detectives, 


recruits
in Math Town go to detective school. 


In
school they learn lots DEFINITIONS. 


Stuff
like: 





STOLEN


If something
is missing and nobody knows 



where it
went. 





ALIBI


If somebody
was away when the thing was stolen, 



they have an
alibi. 





There
are lots of other definitions they have to learn, 


but
that' s enough for us. 





There
are special detective words that are used to tell 


detectives
stuff that is always true. 


Stuff
like: 





When
something gets stolen, 


the
crime of robbery has been committed. 





If
somebody was away when the crime happened, 


they
have an alibi so they didn't do it. 





These
things are called POSTULATES. 





They
are given special names so the detectives can talk about them to
each other 


without
it taking all day. 


The
names of the two we just talked about are: 





The
crime of robbery postulate. 





The
alibi/didn't do it postulate. 





Then
the detective goes to work. 


When
the detective gets to a crime scene 


they
take down some information about what happened. 


Stuff
like: 





1)
Many people saw the missing stuff 


in
the place where it is kept last night at midnight. 





2)
Lots of people saw that the stuff was gone 


this
morning at 8 AM. 





3)
Nobody knows where the stuff is. 





This
information that the detectives are given at the scene 


is
called THE GIVEN INFORMATION (big surprise eh?) 





Now
the fun begins. 


The
detective says "I learned in class that since the stuff is gone 


and
nobody knows where it is, it was STOLEN! 


That
means that the crime of robbery has been committed." 


Then
the detective gets really tricky and figures out: 


"The
stuff was here last night at midnight 


and
gone at 8 AM today. 


That
means the stuff was stolen 


between
midnight and 8 AM!" 


We
call that last bit of detective work a theorem. 


In
general, we could say: 


"The
time something is stolen is between 


the
last time somebody saw it where it was supposed to be 


and
when somebody first noticed it was gone." 





We
can also say that: 





Theorems
are stuff we can figure out 


from
definitions, postulates, and other theorems 


that
we already know. 





People
in Math Town love to name stuff. 


The
detective squad has written down: 





When
We Were Robbed Theorem: 


If
we were robbed, 


the
robbery happened between the last time we saw the stuff 


and
when we find out it's gone. 





We
could even write down what we know so far 


as
a formal Math Town proof: 





GIVEN
INFORMATION 


1)
The stuff was here at midnight last night. 


2)
The stuff was gone at 8 AM today. 


3)
Nobody knows where the stuff is. 





Prove: 


Our
stuff was stolen between midnight last night 


and
8 AM today. 











Definitions,
Theorems and Postulates 


are
the tools the detective uses to figure out what happened. 





They
work any place the detective goes. 





As
time goes by, and the detectives figure out more stuff, 


they
add more Theorems to their crime solving tool kit. 





Once
a detective figures out a new theorem, 


all
the detectives can use it just like the postulates, 


definitions
and other theorems they already have 


to
figure out even more theorems 


and
solve more crimes. 





To
be good at their job, 


a
detective needs a lot of tools. 





That
means having a big list 


of
theorems, postulates and definitions to use. 





The
very best detectives don't even need to read their lists. 





They
use them so often 


that
they know them all from memory. 





The
given information are the facts of a case. 


Things
like what was stolen, when we noticed it was gone, 


and
stuff like that. 


The
given information goes with one particular case. 





Theorems,
postulates and definitions 


work
with every case. 





copyright 2005 Bruce Kirkpatrick 
