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%TCIDATA{<META NAME="Title" CONTENT="Reference/Mathematics/Triangles">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Thu Feb 13 02:43:19 1997}
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\begin{document}


\section{Triangles\label{Triangles}}

\begin{definition}
If $A$, $B$, and $C$ are three points not all on one line (not \label%
{Collinear}\textbf{collinear}), the set of points of the segments $AB$, $BC$%
, $CA$ is a\textbf{\ triangle}. The points $A,$ $B$, and $C$ are \label%
{Vertices}\textbf{vertices} of the triangle and the segments $AB$, $AC$, and 
$BC$ are\label{Sides} \textbf{sides} of the triangle.
\end{definition}

\begin{itemize}
\item  The sum of the measures of the \label{Interior angles}\textbf{%
interior angles} of a triangle is $180\unit{\UNICODE{0xb0}}$.
\end{itemize}

\begin{definition}
A triangle is\label{Acute triangle} \textbf{acute} if all angles are less
than $90\unit{\UNICODE{0xb0}}$,\label{Obtuse triangle} \textbf{obtuse} if
one angle is greater than $90\unit{\UNICODE{0xb0}}$, and a\label%
{Right triangle} \textbf{right triangle} if one angle equals $90\unit{%
\UNICODE{0xb0}}$.
\end{definition}

\subsubsection{Equilateral Triangles\label{Equilateral triangle}}

\begin{definition}
A triangle is \textbf{equilateral} if all sides are of equal length.
\end{definition}

\begin{itemize}
\item  A triangle is equilateral if and only if the three angles are equal,
in which case all angles are $60\unit{\UNICODE{0xb0}}$.\FRAME{dtbpF}{2.0522in%
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\end{itemize}

\subsubsection{Isosceles Triangles\label{Isosceles triangle}}

\begin{definition}
A triangle is \textbf{isosceles} if two of its sides are congruent (of equal
length). The \label{Base angles of isosceles triangle}\textbf{base angles}
of the triangle are those opposite the congruent sides.
\end{definition}

\begin{itemize}
\item  In every isosceles triangle, the two base angles are equal.
\end{itemize}

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\subsubsection{Medians}

\begin{definition}
A\label{Median} \textbf{median} of a triangle is a line segment that has as
its end points a vertex of the triangle and the midpoint of the opposite
side of the triangle.
\end{definition}

\begin{itemize}
\item  The three medians of a triangle intersect in a common point, called
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\item  Any two medians of a triangle intersect at a point that is two-thirds
of the distance from any vertex to the midpoint of the opposite side.
\end{itemize}

\subsubsection{Perpendicular Bisectors of the Sides}

\begin{itemize}
\item  The three \label{Perpendicular bisectors}\textbf{perpendicular
bisectors} of the sides of a triangle intersect in a common point, called
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\end{itemize}

\subsubsection{Angle Bisectors\label{Angle bisectors}}

\begin{itemize}
\item  The three bisectors of the interior angles of a triangle meet at a
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\item  Each angle bisector of a triangle, if sufficiently extended, meets at
a common point (called the \label{Excenter}\textbf{excenter}) with the
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\end{itemize}

\subsubsection{Altitudes\label{Altitude}}

\begin{definition}
An \textbf{altitude} of a triangle is a perpendicular line segment from a
vertex of a triangle to the opposite side (extended if necessary).
\end{definition}

\begin{itemize}
\item  The lines containing the three altitudes of a triangle meet at a
common point, called the\label{Orthocenter} \textbf{orthocenter} of the
triangle.
\end{itemize}

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\subsubsection{Ratio and Proportion\label{Ratio and proportion}}

\begin{itemize}
\item  If $a:b=c:d$ or $\dfrac{a}{b}=\dfrac{c}{d}$, then $ad=bd$ and $\dfrac{%
a}{c}=\dfrac{b}{d}$.\medskip

\item  If $\dfrac{a}{b}=\dfrac{c}{d}=\dfrac{e}{f}=\cdots =\dfrac{g}{h}=k$,
then $k=\dfrac{a+c+e+\cdots +g}{b+d+f+\cdots +h}=\dfrac{pa+qc+re+\cdots +sg}{%
pb+qd+rf+\cdots +sh}$
\end{itemize}

\subsubsection{Similar Triangles\label{Similar triangles}}

\begin{definition}
If $AB$ and $CD$ are line segments, the \label{Ratio of sides}\textbf{ratio
of }$AB$\textbf{\ to }$CD$ is the length of $AB$ divided by the length of $%
CD $.

\textbf{The sides of }$\Delta ABC$\textbf{\ are proportional to the sides of 
}$\Delta DEF$ if the three ratios---the ratio of $AB$ to $DE$, the ratio of $%
BC$ to $EF$, and the ratio of $CA$ to $FD$---are equal.

Two triangles are \label{Similar}\textbf{similar} to each other if
corresponding angles are congruent and corresponding sides are proportional
(have the same ratio).
\end{definition}

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\begin{theorem}
If the angles of one triangle are congruent, respectively, to the angles of
another triangle, then the triangles are similar.
\end{theorem}

\begin{theorem}
A line that divides two sides of a triangle proportionally is parallel to
the third side.
\end{theorem}

\begin{theorem}
If two sides of one triangle are proportional, respectively, to two sides of
another and the included angles are congruent, then the triangles are
similar.
\end{theorem}

\begin{theorem}
If the sides of one triangle are proportional to the sides of another
triangle, then the triangles are similar.
\end{theorem}

\subsection{\ Related Reference Topics}

{\small \hyperref{Circles and Triangles}{}{}{GE04_04.tex}}

{\small \hyperref{Geometry}{}{}{GE_contents.tex}}

{\small \hyperref{Polygons}{}{}{GE04_02.tex}}

{\small \hyperref{Regular Polygon}{}{}{GE01_05.tex}}\hfill

\subsection{\ Related Scientific Notebook Topics}

{\small \hyperref{Solution of Triangles}{}{}{../../Help/DMIndex.tex#Solution
of triangles}}

{\small \hyperref{Trigonometry}{}{}{../../Help/DMIndex.tex#Trigonometry}}%
\hfill 

\noindent \hrulefill

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\noindent \hrulefill

\end{document}
