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\title{General Mathematics and CPS II}
\author{Exercise 6}
\date{February 20, 2015}
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\begin{enumerate}

\item (Ivanov, p.\ 34, Problem 6.)  A billiard ball sits against the
cushion of a billiard table.  In which direction should the ball be
cued so that after bouncing off the three other sides of the table, it
returns to its point of departure?

\item (Ivanov, p.\ 34, Problem 9.)  Consider two squares situated in
the same plane.  Join any corner of one square to any corner of the
other by means of a line segment, and then, proceeding in the same
direction around both squares, the next corner to the next, and so on.

Prove that the midpoints of the four line segments so constructed are
also the vertices of a square.

\item (Ivanov, p.\ 35, Exercise.)  Prove that the composite of two
axial symmetries (an \emph{axial symmetry} is a reflection about a
line, the \emph{axis}) with intersecting axes of symmetry is a
rotation about the point of intersection of these axes through an
angle equal to twice the angle between them.


% \item A \emph{Euclidean transformation} $F$ or, in short, a
% \emph{motion} of the plane is a map which preserves the Euclidean
% distance between points, i.e.
% \[
%   \lvert F(a) - F(b) \rvert = \lvert a - b \rvert
% \]
% where $\lvert a-b \rvert$ denotes the distance between two arbitrary
% points $a$ and $b$.

% Show that every motion of the plane is bijective.

% \emph{Hint:} To prove that it is surjective, notice that the
% transformation must map lines onto lines and circles onto circles.

\end{enumerate}
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