20sep13

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\maketitle

Check your (written !) answer-hints to the practice problems. Be sure supply the details
of any steps you do not understand. Also, work through all indicated calculations.

\textbf{Problem 1}

\begin{itemize}
\item Substitute into $X=aA+bB+cC$ what is given. Then
$X = 0.3A + bB + 0C$. Because $a+b+c=1$
you can solve for $b = 0.7$.
\end {itemize}

\textbf{Problem 2}

\begin{itemize}
\item There is an easy ways of solving this problem. Using Ceva's theorem, you can
find the value of $\frac{C'-A}{C'-B}$ and solve that for the barycentric
coordinates of $C'$. Be sure you know how to sketch the corresponding figure
also freehand, not with some computer geometry tool.
\item A second solution is to express $(A A')$ and $(B B')$ in parametric form
(be sure to use different letters for the parameter.) Then find $G=(A A')(B B')$
by getting three equations in two unknowns (remember, barycentric coordinates are unique.)
They have a unique solution. Now repeat the process for $C' = (AB)(CG)$.
\item By doing this problem both ways, you are checking your arithmetic, which is easy to make
mistakes in. Which method is less work?
\end{itemize}

\textbf{Problems 3}

\begin{itemize}
\item
The point $X = aA + bB + cC$ must have $a = 0$ and $c=2b$.
Substituting and remembering that $1 = a+b+c$ you can solve the three equations for
a unique answer. Be sure you know how to sketch a believable figure for this. You can
estimate the ratios by sight.
\end{itemize}

\textbf{Problems 4}

\begin{itemize}
\item
Observe that from the hypothesis all four points are collinear. So you can put the real numberson the line and treat these differences as real numbers. Draw a figure for a guide.
\end{itemize}

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