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\centerline{MONT 108N -- Mathematics Through Time}
\centerline{Problem Set 4 -- Euclidean Geometry}
\centerline{{\bf due:} November 5, 2010}
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\noindent
I.  Proposition I.5 in the {\it Elements} proves the 
equality of the base angles in an isosceles triangle
using only Propositions I.1 - I.4.  We indicated the 
main idea of the construction for the proof in class
on 10/27.  Work out a complete
proof of this statement, and give a reason for each 
statement.  Is your chain of reasoning exactly the
same as Euclid's?  (There are correct proofs that are
slightly different, of course.)
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\noindent
II.  In many modern treatments of plane geometry, 
an alternate form of the 5th Postulate is used, which 
looks like this:  {\it Given an infinitely produced 
line $\ell$ and a point $P$ not on ${\ell}$, there is 
exactly one infinitely produced line through $P$ that 
is parallel to ${\ell}$.}  This is certainly simpler
than Euclid's Postulate 5, and it can be seen that 
they are equivalent.
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\item{A)} Which Proposition in Euclid gives a 
construction for this parallel line?
\item{B)} Using this form of the 5th Postulate and 
Propositions I.15 and I.29 of the {\it Elements}, prove that
the sum of the angles in any triangle is equal
to a straight angle (that is, the sum is has 
angle measure $180^\circ$; Euclid did not use the
notion of a numerical measure for angles, though). 
\item{C)}  Deduce that the sum of the interior angles
of a convex polygon with $n$ sides is equal to 
$n - 2$ straight angles.  (This is most easily done
as a {\it proof by mathematical induction} to use the 
modern term.  See me in office hours if you 
don't know what that means!)
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\noindent
III.  A deductive sequence concerning areas giving an
alternative to Euclid's derivation of the area formulas
for parallelograms and triangles and another interesting 
fact.  Start from the assumption that the area of a rectangle
is the product of its dimensions (length times width).
Using {\it only facts from Propositions I.1 - I.33}
and previously proved parts of this problem, prove the following.
(This means, for instance, that once you establish A, you can use
that fact in the proof of B and similarly for the other parts.)
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\item{A)}  The area of a parallelogram is equal to the product of
its base and altitude.
\item{B)}  The area of a triangle is equal to the product of
any side and the altitude on that side.
\item{C)}  The area of a right triangle is equal to one half 
the product of its two legs.
\item{D)} The area of a triangle is equal to half the product
of its perimeter and the radius of its inscribed circle. 
(For this part you will also need to assume the fact that a tangent line
to a circle meets the radius at the intersection point in a 
right angle.  This fact is proved in Euclid only in Proposition 18 of 
Book III.)   
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