\documentclass[12pt]{article}
\usepackage{ammaths}\usepackage{pst-eucl}

\begin{document}

\module{Special lines in a triangle}{1}{AP02}{1 period}

\prereq{Knowledge of the special lines and points in a triangle in French.}

\object{\begin{itemize}
        \item Review the special lines and points in a triangle.
	\item Learn the specific vocabulary.
	\end{itemize}}

\mater{\begin{itemize}
	\item Matching cards with figures and words.
	\item Task sheet for the computer work.
	\end{itemize}}

\modpart{Matching and writing definitions game}{15 mins}

Students are handed out cards with either a word or a picture. They mingle to find their match and then work in pairs to
write a definition of their word. Definitions are then gathered by the teacher to be handed out to the class later.

\modpart{Work with Geogebra}{40 mins}

Students, still working in pairs, discover Geogebra and use it to draw all the special points and lines in a triangle, using appropriate colors.

\pagebreak

\moddocdis{Special lines in a triangle}{1}{AP02}{ICT worksheet}

Geogebra is a free software specifically designed to provide ``Dynamic mathematics for schools''. It is centered on
geometry but can also be used to handle functions, parameters and computations.

Geogebra is very user-friendly, each icon offering a series of options when the user clicks on the small triangle on the
bottom-right corner. Here are just a few instructions and tips. 

\begin{itemize}
\itemb Geogebra is multilingual. When used in the Euro maths sessions, it will always have to be set to English,
following the path {\tt Options>Langue>English (UK)}.
\itemb You can choose to view the Algebra Window, the axes and the grid in the View menu (Affichage in French).
\itemb To move the whole drawing pad, use the mouse while hodling the MAJ or CTRL key.
\itemb To zoom in and out, use the scroll wheel on the mouse.
\end{itemize}

You can download Geogebra and install it free of charge from the following website :

\begin{center}
\tt http://www.geogebra.org/download/install.htm
\end{center}



\begin{enumerate}
\item Use GeoGebra to draw a large triangle $ABC$ and the points  $K$, $J$ and
$I$, respective midpoints of the sides $AB$, $AC$ and
$BC$. Then, use the tools available to draw precisely :
 \begin{enumerate}
 \item the perpendicular bisectors in black ;
 \item the medians in blue ;
 \item the altitudes in green ;
 \item the angular bisectors in grey.
 \end{enumerate}
\item Use these lines to build the following special points :
 \begin{enumerate}
 \item the center of the circumcircle $O$ ;
 \item the centroid or center of gravity $G$ ;
 \item the orthocenter $H$ ;
 \item the center of the incircle $P$.
 \end{enumerate}
\item What do you notice about these points and lines in a right-angled
triangle ?
\item What do you notice about these points and lines in an isosceles
triangle ?
\item What do you notice about these points and lines in an equilateral
triangle ?
\end{enumerate}

\pagebreak

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%GLOSSARY
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\moddocdis{Special lines in a triangle}{1}{AP02}{Glossary}



\begin{multicols}{2}
\subsection*{Perpendicular bisector}

\begin{flushright}
Definition by ..................
\end{flushright}

\subsection*{Center of the circumcircle}
\begin{flushright}
Definition by ..................
\end{flushright}

\columnbreak

\null\vfill
\vspace{1cm}

\begin{center}
\psset{unit=1.5cm}
%Perpendicular bisectors
\begin{pspicture}(-0.5,-1.7)(3.5,2.7)
\psset{PointSymbol=none}
\pstTriangle[PointName=none](0,0.5){A}(2.5,0.5){B}(1.7,2.5){C}
\psset{CodeFig=true}
\pstCircleABC[RightAngleSize=0.2]{A}{B}{C}{O}
\psset{linecolor=black}
\pstTriangle[PointName=none](0,0.5){A}(2.5,0.5){B}(1.7,2.5){C}
\end{pspicture}
\end{center}

%\null\vfill
\end{multicols}

%\bigskip

\hrule

%\bigskip

\begin{multicols}{2}
\subsection*{Altitude}

\begin{flushright}
Definition by ..................
\end{flushright}



\subsection*{Orthocenter}

\begin{flushright}
Definition by ..................
\end{flushright}

\columnbreak

\null\vfill

\begin{center}
\psset{unit=1.5cm}
%Altitudes and orthocenter
\begin{pspicture}(-0.5,-0.5)(3.5,2.2)
\psset{CodeFig=true, PointSymbol=none}
\pstTriangle[PosAngleA=180,PointName=none]{A}(3,0){B}(2,2){C}
\pstProjection[PosAngle=-90, RightAngleSize=0.2,PointName=none]{B}{A}{C}
\pstProjection[RightAngleSize=0.2,PointName=none]{B}{C}{A}
\pstProjection[PosAngle=90,RightAngleSize=0.2,PointName=none]{A}{C}{B}
\pstInterLL[PosAngle=180, PointSymbol=square]{A}{A'}{B}{B'}{H}
\end{pspicture}
\end{center}

\null\vfill

\end{multicols}

%\bigskip

\hrule

%\bigskip

\begin{multicols}{2}

\subsection*{Median}

\begin{flushright}
Definition by ..................
\end{flushright}

\subsection*{Center of gravity}

\begin{flushright}
Definition by ..................
\end{flushright}

\columnbreak

\null\vfill

\begin{center}
\psset{unit=1.5cm}
%Medians
\begin{pspicture}(-0.5,-0.1)(3.5,3)
\psset{PointSymbol=none}
\pstTriangle[PointName=none]{A}(3,0){B}(2,2.5){C}
\pstMiddleAB[PosAngle=-90, SegmentSymbol=pstslash, PointName=none]{A}{B}{K}
\pstMiddleAB[PosAngle=20, SegmentSymbol=pstslashh, PointName=none]{B}{C}{I}
\pstMiddleAB[PosAngle=150, SegmentSymbol=MarkCros, PointName=none]{C}{A}{J}
\pstLineAB[linestyle=dashed]{A}{I}
\pstLineAB[linestyle=dashed]{B}{J}
\pstLineAB[linestyle=dashed]{C}{K}
\pstCGravABC{A}{B}{C}{G}
\end{pspicture}
\end{center}

\null\vfill

\end{multicols}

%\bigskip

%\hrule
\pagebreak
%\bigskip


\begin{multicols}{2}

\subsection*{Angular bisector}

\begin{flushright}
Definition by ..................
\end{flushright}

\subsection*{Center of the inside circle}

\begin{flushright}
Definition by ..................
\end{flushright}

\columnbreak

\null\vfill

\begin{center}
\psset{unit=1.5cm}
%Angular bissectors
\begin{pspicture}(-0.5,-0.5)(3.5,2.7)
\psset{PointSymbol=none}
\pstTriangle[PointName=none]{A}(3,0){B}(2,2.5){C}
\pstBissectBAC[linestyle=dashed, PointName=none]{B}{A}{C}{A'}
\pstBissectBAC[linestyle=dashed, PointName=none]{A}{C}{B}{C'}
\pstBissectBAC[linestyle=dashed, PointName=none]{C}{B}{A}{B'}
\pstMarkAngle[Mark=MarkHash, MarkAngleRadius=0.8]{B}{A}{A'}{}
\pstMarkAngle[Mark=MarkHash, MarkAngleRadius=0.7]{A'}{A}{C}{}
\pstMarkAngle[MarkAngleRadius=0.8]{C}{B}{B'}{}
\pstMarkAngle[MarkAngleRadius=0.9]{B'}{B}{A}{}
\pstMarkAngle[Mark=MarkCros, MarkAngleRadius=0.7]{A}{C}{C'}{}
\pstMarkAngle[Mark=MarkCros, MarkAngleRadius=0.6]{C'}{C}{B}{}
\pstInterLL{C}{C'}{A}{A'}{I}
\pstProjection[PointName=none]{A}{B}{I}[I']
\pstCircleOA{I}{I'}
\end{pspicture}
\end{center}

\null\vfill

\end{multicols}


\bigskip

\hrule

\bigskip


\begin{multicols}{2}

\subsection*{Isosceles triangle}

\begin{flushright}
Definition by ..................
\end{flushright}

\vfill

\columnbreak

\null\vfill

\begin{center}
\psset{unit=1.5cm}
%Isosceles triangle
\begin{pspicture}(-0.5,0)(3.5,2.5)
\pstTriangle[PointSymbol=none,PointName=none](0,0){A}(3,0){B}(1.5,2.5){C}
\psset{SegmentSymbol=MarkCros}\pstSegmentMark{A}{C}\pstSegmentMark{C}{B}
\pstMarkAngle[Mark=MarkHash]{C}{B}{A}{}
\pstMarkAngle[Mark=MarkHash]{B}{A}{C}{}
\end{pspicture}
\end{center}

\null\vfill

\end{multicols}

\bigskip

\hrule

\bigskip


\begin{multicols}{2}

\subsection*{Right-angled triangle}

\begin{flushright}
Definition by ..................
\end{flushright}

\vfill
\columnbreak

\null\vfill

\begin{center}
\psset{unit=1.5cm}
%Right-angled triangle
\begin{pspicture}(-0.7,-0.5)(3.5,2.3)
\pstTriangle[PointSymbol=none,PointName=none](0,0){A}(0,2){B}(3,0){C}
\pstRightAngle{C}{A}{B}
\end{pspicture}
\end{center}

\null\vfill

\end{multicols}


\pagebreak
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%MATCHING CARDS
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\moddoc{Matching cards}

\begin{center}\LARGE
\begin{tabular}{|p{3cm}cp{3cm}|}
\hline
& Perpendicular bisector & \rule{0pt}{25pt}\\[8pt]
\hline
& Altitude & \rule{0pt}{25pt}\\[8pt]
\hline
& Median &\rule{0pt}{25pt}\\[8pt]
\hline
& Angular bisector & \rule{0pt}{25pt}\\[8pt]
\hline
& Center of the circumcircle & \rule{0pt}{25pt}\\[8pt]
\hline
& Orthocenter & \rule{0pt}{25pt}\\[8pt]
\hline
& Center of gravity & \rule{0pt}{25pt}\\[8pt]
\hline
& Center of the inside circle & \rule{0pt}{25pt}\\[8pt]
\hline
& Isosceles triangle &\rule{0pt}{25pt}\\[8pt]
\hline
& Right-angled triangle & \rule{0pt}{25pt}\\[8pt]
\hline
\end{tabular}
\end{center}

\pagebreak
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%LES FIGURES
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
\fbox{\psset{xunit=0.7cm,yunit=0.7cm,algebraic=true,dotstyle=*,dotsize=3pt 0,linewidth=0.8pt,arrowsize=3pt
2,arrowinset=0.25}
\begin{pspicture*}(-1.98,-4.06)(11.66,6.02)
\psline(0,3)(4,0)
\psline(4,0)(10,4)
\psline(10,4)(0,3)
\psplot[linestyle=dashed]{-1.98}{11.66}{(-3.5--4*x)/3}%Perpendicular bisector
\psdots(0,3)
\psdots(4,0)
\psdots(10,4)
\psdots(2,1.5)
\psdots(7,2)
\psdots(5,3.5)
\end{pspicture*}}

\fbox{\psset{xunit=0.7cm,yunit=0.7cm,algebraic=true,dotstyle=*,dotsize=3pt 0,linewidth=0.8pt,arrowsize=3pt
2,arrowinset=0.25}
\begin{pspicture*}(-1.98,-4.06)(11.66,6.02)
\psline(0,3)(4,0)
\psline(4,0)(10,4)
\psline(10,4)(0,3)
\psplot[linestyle=dashed]{-1.98}{11.66}{(-28--4*x)/3}%Altitude
\psdots(0,3)
\psdots(4,0)
\psdots(10,4)
\psdots(2,1.5)
\psdots(7,2)
\psdots(5,3.5)
\end{pspicture*}}


\fbox{\psset{xunit=0.7cm,yunit=0.7cm,algebraic=true,dotstyle=*,dotsize=3pt 0,linewidth=0.8pt,arrowsize=3pt
2,arrowinset=0.25}
\begin{pspicture*}(-1.98,-4.06)(11.66,6.02)
\psline(0,3)(4,0)
\psline(4,0)(10,4)
\psline(10,4)(0,3)
\psplot[linestyle=dashed]{-1.98}{11.66}{(-14--3.5*x)/1}%Median
\psdots(0,3)
\psdots(4,0)
\psdots(10,4)
\psdots(2,1.5)
\psdots(7,2)
\psdots(5,3.5)
\end{pspicture*}}

\fbox{\psset{xunit=0.7cm,yunit=0.7cm,algebraic=true,dotstyle=*,dotsize=3pt 0,linewidth=0.8pt,arrowsize=3pt
2,arrowinset=0.25}
\begin{pspicture*}(-1.98,-4.06)(11.66,6.02)
\psline(0,3)(4,0)
\psline(4,0)(10,4)
\psline(10,4)(0,3)
\psplot[linestyle=dashed]{-1.98}{11.66}{(--2.89-0.27*x)/0.96}%Angular bisector
\psdots(0,3)
\psdots(4,0)
\psdots(10,4)
\psdots(2,1.5)
\psdots(7,2)
\psdots(5,3.5)
\end{pspicture*}}


\fbox{\psset{xunit=0.7cm,yunit=0.7cm,algebraic=true,dotstyle=*,dotsize=3pt 0,linewidth=0.8pt,arrowsize=3pt
2,arrowinset=0.25}
\begin{pspicture*}(-1.98,-4.06)(11.66,6.02)
\psline(0,3)(4,0)
\psline(4,0)(10,4)
\psline(10,4)(0,3)
\psplot[linestyle=dashed]{-1.98}{11.66}{(-3.5--4*x)/3}%Perpendicular bisector
\psplot[linestyle=dashed]{-1.98}{11.66}{(-50--6*x)/-4}
\psplot[linestyle=dashed]{-1.98}{11.66}{(--53.5-10*x)/1}
\psdots(0,3)
\psdots(4,0)
\psdots(10,4)
\psdots(2,1.5)
\psdots(7,2)
\psdots(5,3.5)
\end{pspicture*}}


\fbox{\psset{xunit=0.7cm,yunit=0.7cm,algebraic=true,dotstyle=*,dotsize=3pt 0,linewidth=0.8pt,arrowsize=3pt
2,arrowinset=0.25}
\begin{pspicture*}(-1.98,-4.06)(11.66,6.02)
\psline(0,3)(4,0)
\psline(4,0)(10,4)
\psline(10,4)(0,3)
\psplot[linestyle=dashed]{-1.98}{11.66}{(-28--4*x)/3}%Altitude
\psplot[linestyle=dashed]{-1.98}{11.66}{(-12--6*x)/-4}
\psplot[linestyle=dashed]{-1.98}{11.66}{(--40-10*x)/1}
\psdots(0,3)
\psdots(4,0)
\psdots(10,4)
\psdots(2,1.5)
\psdots(7,2)
\psdots(5,3.5)
\end{pspicture*}}


\fbox{\psset{xunit=0.7cm,yunit=0.7cm,algebraic=true,dotstyle=*,dotsize=3pt 0,linewidth=0.8pt,arrowsize=3pt
2,arrowinset=0.25}
\begin{pspicture*}(-1.98,-4.06)(11.66,6.02)
\psline(0,3)(4,0)
\psline(4,0)(10,4)
\psline(10,4)(0,3)
\psplot[linestyle=dashed]{-1.98}{11.66}{(-14--3.5*x)/1}%Median
\psplot[linestyle=dashed]{-1.98}{11.66}{(--21-1*x)/7}
\psplot[linestyle=dashed]{-1.98}{11.66}{(-7-2.5*x)/-8}
\psdots(0,3)
\psdots(4,0)
\psdots(10,4)
\psdots(2,1.5)
\psdots(7,2)
\psdots(5,3.5)
\end{pspicture*}}


\fbox{\psset{xunit=0.7cm,yunit=0.7cm,algebraic=true,dotstyle=*,dotsize=3pt 0,linewidth=0.8pt,arrowsize=3pt
2,arrowinset=0.25}
\begin{pspicture*}(-1.98,-4.06)(11.66,6.02)
\psline(0,3)(4,0)
\psline(4,0)(10,4)
\psline(10,4)(0,3)
\psplot[linestyle=dashed]{-1.98}{11.66}{(--2.89-0.27*x)/0.96}%Angular bisector
\psplot[linestyle=dashed]{-1.98}{11.66}{(-0.39-0.34*x)/-0.94}
\psplot[linestyle=dashed]{-1.98}{11.66}{(-4--1*x)/0.03}
\psdots(0,3)
\psdots(4,0)
\psdots(10,4)
\psdots(2,1.5)
\psdots(7,2)
\psdots(5,3.5)
\end{pspicture*}}


\fbox{\psset{xunit=1cm,yunit=1cm,algebraic=true,dotstyle=*,dotsize=3pt 0,linewidth=0.8pt,arrowsize=3pt
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\psline(3,4)(1,-2)
\psline(2.06,0.98)(1.94,1.02)
\psline(1,-2)(7,0)
\psline(3.98,-0.94)(4.02,-1.06)
\psline(7,0)(3,4)
\psdots(3,4)
\psdots(1,-2)
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}




\end{center}

\pagebreak
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%LESSON
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\moddocdis{Special lines in a triangle}{1}{03}{Glossary}

All of this definitions come from \url{http://en.wikipedia.org/wiki/Triangle}

\begin{multicols}{2}
\subsection*{Perpendicular bisector}
A perpendicular bisector of a side of a triangle is a straight line passing through the midpoint of the side and being
perpendicular to it.\\
The perpendicular bisector of a segment also has the property that each of its points is equidistant from the endpoints
of the segment.

\subsection*{Center of the circumcircle}
The three perpendicular bisectors meet in a single point, the triangle's circumcenter; this point is the center of the
circumcircle, the circle passing through all three vertices.

\columnbreak

\null\vfill
\vspace{1cm}

\begin{center}
\psset{unit=1.5cm}
%Perpendicular bisectors
\begin{pspicture}(-0.5,-1.7)(3.5,2.7)
\psset{PointSymbol=none}
\pstTriangle[PointName=none](0,0.5){A}(2.5,0.5){B}(1.7,2.5){C}
\psset{CodeFig=true}
\pstCircleABC[RightAngleSize=0.2]{A}{B}{C}{O}
\psset{linecolor=black}
\pstTriangle[PointName=none](0,0.5){A}(2.5,0.5){B}(1.7,2.5){C}
\end{pspicture}
\end{center}

%\null\vfill
\end{multicols}

%\bigskip

\hrule

%\bigskip

\begin{multicols}{2}
\subsection*{Altitude}
An altitude of a triangle is a straight line passing through a vertex and perpendicular to (i.e. forming a right angle
with) the opposite side or an extension of the opposite side.




\subsection*{Orthocenter}
The three altitudes intersect in a single point, called the orthocenter of the triangle.

\columnbreak

\null\vfill

\begin{center}
\psset{unit=1.5cm}
%Altitudes and orthocenter
\begin{pspicture}(-0.5,-0.5)(3.5,2.2)
\psset{CodeFig=true, PointSymbol=none}
\pstTriangle[PosAngleA=180,PointName=none]{A}(3,0){B}(2,2){C}
\pstProjection[PosAngle=-90, RightAngleSize=0.2,PointName=none]{B}{A}{C}
\pstProjection[RightAngleSize=0.2,PointName=none]{B}{C}{A}
\pstProjection[PosAngle=90,RightAngleSize=0.2,PointName=none]{A}{C}{B}
\pstInterLL[PosAngle=180, PointSymbol=square]{A}{A'}{B}{B'}{H}
\end{pspicture}
\end{center}

\null\vfill

\end{multicols}

%\bigskip

\hrule

%\bigskip

\begin{multicols}{2}

\subsection*{Median}
A median of a triangle is a straight line passing through a vertex and the midpoint of the opposite side, and divides
the triangle into two equal areas.


\subsection*{Center of gravity}
The three medians intersect in a single point, the center of gravity of the triangle.
\columnbreak

\null\vfill

\begin{center}
\psset{unit=1.5cm}
%Medians
\begin{pspicture}(-0.5,-0.1)(3.5,3)
\psset{PointSymbol=none}
\pstTriangle[PointName=none]{A}(3,0){B}(2,2.5){C}
\pstMiddleAB[PosAngle=-90, SegmentSymbol=pstslash, PointName=none]{A}{B}{K}
\pstMiddleAB[PosAngle=20, SegmentSymbol=pstslashh, PointName=none]{B}{C}{I}
\pstMiddleAB[PosAngle=150, SegmentSymbol=MarkCros, PointName=none]{C}{A}{J}
\pstLineAB[linestyle=dashed]{A}{I}
\pstLineAB[linestyle=dashed]{B}{J}
\pstLineAB[linestyle=dashed]{C}{K}
\pstCGravABC{A}{B}{C}{G}
\end{pspicture}
\end{center}

\null\vfill

\end{multicols}

%\bigskip

%\hrule
\pagebreak
%\bigskip


\begin{multicols}{2}

\subsection*{Angular bisector}
An angular bisector  of a triangle is a straight line through a vertex which cuts the corresponding angle in half. 



\subsection*{Center of the inside circle}
The three angular bisectors intersect in a single point, the incenter, the center of the triangle's incircle. 
\columnbreak

\null\vfill

\begin{center}
\psset{unit=1.5cm}
%Angular bissectors
\begin{pspicture}(-0.5,-0.5)(3.5,2.7)
\psset{PointSymbol=none}
\pstTriangle[PointName=none]{A}(3,0){B}(2,2.5){C}
\pstBissectBAC[linestyle=dashed, PointName=none]{B}{A}{C}{A'}
\pstBissectBAC[linestyle=dashed, PointName=none]{A}{C}{B}{C'}
\pstBissectBAC[linestyle=dashed, PointName=none]{C}{B}{A}{B'}
\pstMarkAngle[Mark=MarkHash, MarkAngleRadius=0.8]{B}{A}{A'}{}
\pstMarkAngle[Mark=MarkHash, MarkAngleRadius=0.7]{A'}{A}{C}{}
\pstMarkAngle[MarkAngleRadius=0.8]{C}{B}{B'}{}
\pstMarkAngle[MarkAngleRadius=0.9]{B'}{B}{A}{}
\pstMarkAngle[Mark=MarkCros, MarkAngleRadius=0.7]{A}{C}{C'}{}
\pstMarkAngle[Mark=MarkCros, MarkAngleRadius=0.6]{C'}{C}{B}{}
\pstInterLL{C}{C'}{A}{A'}{I}
\pstProjection[PointName=none]{A}{B}{I}[I']
\pstCircleOA{I}{I'}
\end{pspicture}
\end{center}

\null\vfill

\end{multicols}


\bigskip

\hrule

\bigskip


\begin{multicols}{2}

\subsection*{Isosceles triangle}

In an isosceles triangle, two sides are equal in length. An isosceles triangle also has two angles of the same measure.
\vfill

\columnbreak

\null\vfill

\begin{center}
\psset{unit=1.5cm}
%Isosceles triangle
\begin{pspicture}(-0.5,0)(3.5,2.5)
\pstTriangle[PointSymbol=none,PointName=none](0,0){A}(3,0){B}(1.5,2.5){C}
\psset{SegmentSymbol=MarkCros}\pstSegmentMark{A}{C}\pstSegmentMark{C}{B}
\pstMarkAngle[Mark=MarkHash]{C}{B}{A}{}
\pstMarkAngle[Mark=MarkHash]{B}{A}{C}{}
\end{pspicture}
\end{center}

\null\vfill

\end{multicols}

\bigskip

\hrule

\bigskip


\begin{multicols}{2}

\subsection*{Right-angled triangle}
A right triangle (or right-angled triangle, formerly called a rectangled triangle) has one of its interior angles measuring 90$^o$ (a right angle).\\
The side opposite to the right angle is the hypotenuse; it is the longest side of the right triangle. The other two sides are called the legs of the triangle.
\vfill
\columnbreak

\null\vfill

\begin{center}
\psset{unit=1.5cm}
%Right-angled triangle
\begin{pspicture}(-0.7,-0.5)(3.5,2.3)
\pstTriangle[PointSymbol=none,PointName=none](0,0){A}(0,2){B}(3,0){C}
\pstRightAngle{C}{A}{B}
\end{pspicture}
\end{center}

\null\vfill

\end{multicols}
\end{document}
