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%TCIDATA{Created=Thu Jul 10 12:07:18 1997}
%TCIDATA{LastRevised=Mon Dec 10 10:28:13 2001}
%TCIDATA{Language=American English}

\input tcilatex
\begin{document}

\title{Inscribed and Circumscribed Circles}
\date{December 2001}
\author{Ellen Maycock}
\maketitle

How do we construct the inscribed and circumscribed circles of a triangle in
various geometries? Do similar constructions hold in Euclidean, spherical
and hyperbolic geometry? In this lab, we will investigate these
constructions and see some of the special features of these circles.

\section{Before the lab}

This section should be done as homework before the laboratory period.

Remember that the {\bf circumscribed circle} of a triangle is the circle
that goes through the 3 vertices of the triangle. The center of this
circumscribed circle is called the {\bf circumcenter}. The {\bf inscribed
circle} of a triangle is the circle that touches each side of the triangle
exactly once. The center of this circle is called the {\bf incenter}.

Let's begin with a compass and straightedge construction in Euclidean
geometry. You may use pencil and paper or a dynamic software such as {\it %
Geometer's Sketchpad} to do your constructions.

\begin{enumerate}
\item[1.]  Draw an arbitrary triangle $ABC.$

\item[2.]  Construct the angle bisectors of each angle of the triangle. If
you are using a dynamic program, convince yourself that the angle bisectors
always meet in one point by moving the vertices around. Which circle would
have this point as a center? Why?

\item[3.]  Construct the perpendicular bisectors of each side of the
triangle. As before, convince yourself that they always meet in one point.
Which circle would have this point as a center? Why?
\end{enumerate}

\section{In the Lab}

You will work through this section with your partner during the laboratory
period.

You will use the {\it L\'{e}n\'{a}rt Sphere} to consider these constructions
in spherical (elliptic) geometry and the dynamic program {\it %
Poincar\'{e}Draw} to investigate them in hyperbolic geometry.

\subsection{Spherical Geometry}

You will need to answer the following questions by working on the {\it %
L\'{e}n\'{a}rt Sphere}.

\begin{enumerate}
\item[4.]  Draw a triangle $ABC$ on the {\it L\'{e}n\'{a}rt Sphere}. Then
construct the polar triangle $A^{*}B^{*}C^{*}$ associated to triangle $ABC$.
(See below if you are not familiar with the polar triangle).

\item[5.]  Draw the angle bisectors of the original triangle $ABC$. Do they
intersect in a point? More than one point? No points? Be sure to consider
some extreme cases, such as when all the vertices lie on one great circle.
Do the angle bisectors of triangle $ABC$ relate to triangle $A^{*}B^{*}C^{*}$
? How?

\item[6.]  For the cases when you found a common point of intersection, use
this point as the center to draw a special circle for the triangle $ABC$. Is
this the same construction you used in Euclidean geometry? Also use this
point to draw a special circle for triangle $A^{*}B^{*}C^{*}.$

\item[7.]  Now draw the perpendicular bisectors for the sides of triangle $%
ABC$. Do they intersect in a point? More than one point? Be sure to consider
some extreme cases. For the cases when you found a common point of
intersection, use the point to construct a special circle for triangle $ABC$%
. Is this the same construction as in Euclidean geometry? Do the
perpendicular bisectors of the sides of triangle $ABC$ relate to the
triangle $A^{*}B^{*}C^{*}$ ?

\item[8.]  Write a conjecture based on your work in this section. Explain in
your own words why you think your conjecture is true.
\end{enumerate}

\subsection{Hyperbolic geometry}

You will need to answer the following questions by working in the computer
lab, using the program {\it Poincar\'eDraw}.

\begin{enumerate}
\item[9.]  Plot 3 points on the screen, and construct the triangle $abc.$
Then construct the angle bisectors of each angle. Do they intersect in one
point? Move the triangle around the screen to consider different
possibilities, making sure you include some extreme cases.

\item[10.]  Try to construct an inscribed circle based on your work in the
problem above. Does the construction agree with the construction in
Euclidean geometry? Does the construction always work? Why or why not?

\item[11.]  Now construct the perpendicular bisectors of each side of
triangle $abc$. Do they intersect in a point? If so, draw a circumscribed
circle for triangle $abc$ using this point as a center. Use the dynamic
property of the program to see if the construction holds for all triangles.

\item[12.]  What problems did you run into trying to answer question 11? Can
you make a conjecture based on the data? What possibilities can occur with
the perpendicular bisectors? When will you be able to construct a
circumscribed circle?
\end{enumerate}

\subsection{Further work}

This section should be done as homework after the laboratory period.

\begin{enumerate}
\item[13.]  Write a careful proof that the perpendicular bisectors of the
sides of a triangle intersect in a point that is the center of the
circumscribed circle in Euclidean geometry.

\item[14.]  Write a careful proof that the angle bisectors of a triangle
intersect in a point that is the center of the inscribed circle in Euclidean
geometry.
\end{enumerate}

\end{document}
