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\centerline{MONT 104Q -- Mathematical Journeys: From Known To Unknown}
\centerline{Working Within Euclid's System -- September 30, 2015}
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{\it Background}
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By this time we have covered up to Proposition 16 in class, and you have read through Proposition 22 in 
Book I of the {\it Elements}.  To recap, recall that we know the Common Notions and Postulates, 
a number of basic constructions, as well as the following facts:
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\item{} Proposition 12: Given a line and a point not on the line we can construct a perpendicular
from the point to the line.  
\item{} Proposition 15: Vertical angles at the intersection of two straight lines are equal
\item{} Proposition 16: An exterior angle of a triangle is greater than either nonadjacent 
interior angle
\item{} Proposition 17: The sum of each pair of interior angles in a triangle is less than 
two right angles (i.e. less than $180^\circ$ for us).
\item{} Proposition 18: If in a triangle one side is greater than another side, then the angle
opposite the greater side is greater than the angle opposite the other side.  
\item{} Proposition 19:  If in a triangle one interior angle is greater than another interior
angle, then the side opposite the greater angle is greater than the side opposite the other angle.
\item{} Proposition 20:  (the ``{\it triangle inequality}'') The sum of any two sides of a triangle is greater than the other side.    
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Even though Euclid never quite states this, he uses and we will also use an additional ``common notion''
today:  
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\item{} (``CN 6'') {\it If one thing is greater than a second and equals are added to both, then 
the first sum is greater than the second sum.}
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{\it Side trips on our main journey}
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Using {\it only} (I repeat, {\it only}!) what we know at this point, prove the 
following, with justifications for every step.  
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\item{A)} (A ``warm-up'') Using Proposition 16, prove Proposition 17.  (Hint:  Let the triangle be
$\Delta ABC$.  ``Without loss of generality'' we can consider the two interior angles at $A$ and $B$.  
Extend the side $CB$ to $CD$ and apply Proposition 16 to the exterior angle $<DBA$.)
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\item{B)} Let $\Delta ABC$ be an equilateral triangle and let $D$ be a point on the side 
$BC$ different from $B,C$.  Prove that the line segment $AD$ is shorter than the sides of the triangle. 
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\item{C)}  Prove the following important fact:  Given a line and a point $P$ not on the line, 
the perpendicular from $P$ to the line (the result of the construction from Proposition 12) is the 
{\it shortest} line segment joining $P$ to any point on the line.  ({\it Note}: You will probably be
tempted to use the Pythagorean Theorem here.  We have not proved that!)
\bye
