\chapter{Geometry}   
   
\section{Plane geometry}   
   
\Citation{There is no `royal road' to geometry.\\ 
  \textsc{Euclid}\,\footnotemark (4th--3rd c. \textsc{bc}) Greek mathematician\\ 
  Adressed to Ptolemy I} \footnotetext{\textsc{Euclid} \pron{£hj£Ukl£id} taught in Alexandria \pron{£b£al£i£hgz£Andr£Q} 
  circa \pron{£hs£Ek£e} 300 \textsc{bc}, and was probably the founder of its mathematical school. His chief extant work 
  is the 13-volume \emph{Elements}, which became the most widely known mathematical book of classical antiquity.} 
   
The plane \pron{pl£qn} geometry \pron{d£j£i£h£om£etr£i} deals with figures such as points \pron{p£rnt}, lines 
\pron{l£pn}, triangles \pron{£htr£p£a£ngl}, circles \pron{£hs£Ekl}, quadrilaterals 
\pron{£bkw£o\-dr£i\-£hl£a\-t(£e)r(£e)l}\dots which lie in a plane. You can think of a plane as a flat surface which 
extends infinitely \pron{£h£inf£in£etl£i} in all direction. One can represent a plane by a piece of paper or a blackboard. 
   
\subsection{Points and lines}   
   
Given two distinct points $A$ and $B$ in the plane, there is one and only one line which goes or passes through those 
points. We denote that line by $(AB)$.  The point $A$ is \emph{on the line} $(AB)$. We can denote that fact by $A \in 
(AB)$. 
If points are on a same line, they are said to be \nuvo{collinear} \pron{k£o£hl£inj£e£Z}.%   
\index{on}\index{line}\index{point}\index{$(AB)$}\index{$\in$} 
   
The line segment or segment between $A$ and $B$ is the set consisting of $A$, $B$ and all the points on the line $(AB)$ 
lying between $A$ and $B$. We denote this segment by $[AB]$.  The length of the segment $[AB]$ is also the distance from 
$A$ to $B$. It is denoted by $AB$. 
   
The \nuvo{midpoint} \pron{£hm£idp£rnt} $I$ of the segment $[AB]$ is the point such that $I$ is on $[AB]$ 
and $AI = IB$.%   
\index{distance}\index{distance@$AB$} \index{segment}\index{$[AB]$} 
   
Let \Li and \Lip be two lines. If $\Li = \Lip$ or if \Li and \Lip have no common point, \Li and \Lip are \nuvo{parallel} 
\pron{£hp£ar£elel}. If \Li and \Lip are not parallel, they meet in exactly one point.  They cut each other.  If $A$ is 
the common point of \Li and \Lip, we can say that \Li and \Lip intersect \pron{£b£int£e£hsekt} at $A$.  $A$ is their 
\emph{intersection point} \pron{£b£int£e£hsek£sn}.\index{intersection}\index{point!intersection} 
   
Let $A$ be a point on the line \Li. $A$ defines on \Li two \emph{rays} \pron{r£q} or \emph{half lines} starting from 
$A$.  If $B$ and $C$ are on \Li such that $A$ is on $[BC]$, we can denote those rays by $[AB)$ and $[AC)$ respectively. 
$A$ is called the \nuvo{vertex} \pron{£hv£Eteks} of the rays $[AB)$ and $[AC)$.%   
\index{ray}\index{half line}\index{vertex!of a ray}\index{$[AB)$} 
   
\begin{fgre}
  \IG{demidroite}
\end{fgre}
   
\subsection{Angles}   
   
Let $A$, $B$ and $C$ be three collinear points such that $B \in [AC]$.% correction du 17/11/2000   
They define a \emph{straight angle} \pron{streit £a£ngl} viz.\,\footnote{viz. \pron{v£iz} or \pron{£hn£qml£i}: namely; 
  that is to say; in other words. (abbr. of the Latin \textsc{videlicet}, \emph{z} being a medieval Latin symbol for 
  abbreviation 
of \emph{-et})} $\angle ABC$ `angle $ABC$'. The angle $\angle ABA$ is said to be \emph{full}.%   
\index{angle!straight}\index{viz.\cfnote}\index{angle!full} 
   
Two rays with the same vertex separate the plane into two regions. Each one of these regions together with the rays is 
called an \nuvo{angle} determined by the rays, \parex\ $\angle ABC$ is determined by the rays $[BA)$ and $[BC)$. Those 
rays are called the \emph{legs} \pron{leg} of the angle $ABC$.\index{leg} 
   
An angle can be \nuvo{acute} \pron{£e£hkj£Ut}, \nuvo{right} \pron{r£pt} or \nuvo{obtuse} \pron{£eb£htj£Us}.%   
\index{angle!acute}\index{angle!right}\index{angle!obtuse} 
   
\begin{fgre}
  \IG{angles_aod}
\end{fgre}

Two angles are \nuvo{adjacent} \pron{£e£hd£j£qs£ent} if they have a ray in common.
   
We can measure angles with \emph{degrees} \pron{d£i£hgr£I}.  A right angle is an angle of 90\degre. A straight angle has 
180\degre.  If the sum of two angles is equal to 90\degre, those angles are said to be \nuvo{complementary} 
\pron{£bk£ompl£i£hment£er£i}.  If their sum is equal to 180\degre, they are said to be \nuvo{supplementary} 
\pron{£bs£vpl£i£hment£er£i}.  To measure an angle we use a \nuvo{protractor} \pron{pr£e£htr£akt£e£Z}.\index{degree} 
   
We define the \emph{angle bisector} to be the ray which divides the angle into two adjacent angles having the same 
measure.\index{bisector!angle}\index{bisector} 
   
We define two lines to be \nuvo{perpendicular} \pron{£bp£Epen£hd£ikj£ul£e£Z} if they intersect, and if the angle between 
the lines is a right angle. The \nuvo{mid-perpendicular} or \emph{perpendicular bisector} of segment $[AB]$ is the line 
which passes through the midpoint of $[AB]$ and which is perpendicular to $(AB)$.\index{bisector!perpendicular} 
   
On the figure below, $\angle CAB$ and $\angle BAD$ are adjacent and supplementary; $\angle CAB$ and $\angle EAD$ are 
said to be \nuvo{vertical} \pron{£hv£Et£ikl}; $\angle CAB$ and $\angle FEA$ are said to be \nuvo{corresponding} 
\pron{£bk£or£is£hp£ond£i£n}; $\angle CAB$ and $\angle GEH$ are \nuvo{alternate} \pron{£Ol£ht£En£et}. 
   
\begin{fgre}
  \IG{alt_int}
\end{fgre}
   
Line $(AE)$ is called a \nuvo{transversal} \pron{tr£anz£hv£Esl} or \nuvo{traverse} \pron{£htr£av£es} for it intersects 
two (or more) lines. 
   
\subsection{Polygons}   
A \nuvo{polygon} \pron{£hp£ol£ig£en} is a closed plane figure bounded by three or more straight line segments that 
terminate in pairs at the same number of vertices, and do not intersect other than at their vertices. A polygon is 
\nuvo{regular} \pron{£hregj£ul£e£Z} if it has all its sides and all its angles equal. Specific polygons have names that 
indicate the number of sides such as triangle, quadrilateral, \nuvo{pentagon} \pron{£hpent£eg£en} ---~five sides~---, 
\nuvo{hexagon} \pron{£hheks£eg£en} ---~six sides~---, and so on.\index{polygon}\index{regular}\index{pentagon}\index{hexagon} 
   
A polygon is \nuvo{convex} \pron{£hk£onveks} if it has no interior angle greater than $180$\degre. On the figure below 
$ADBEF$ is convex whereas $ACBEF$ is not. A polygon which is not convex is said to be \nuvo{concave} \pron{£hk£o£nk£qv}. 
   
% corrections faites dans la figure (17/11/2000)   
\begin{fgre}
  \IG{conv_conc}
\end{fgre}
   
\subsection{Triangles}   
   
A triangle is a three-sided plane figure. Three non-collinear points determine a triangle. If $A$, $B$ and $C$ are not 
collinear, they are the \emph{vertices} \pron{£hv£Et£is£Iz} (plural of `vertex' \pron{£hv£Eteks}) of the triangle $ABC$. 
Triangle $ABC$ can be denoted by $\triangle ABC$.\index{vertices|see{vertex}}\index{vertex}\index{vertex!of a figure} 
   
Triangles can be classified by the relative \pron{£hrel£et£iv} lengths of their sides. A \nuvo{scalene} 
\pron{£hsk£ql£In} triangle has no sides of equal length; an \nuvo{isosceles} \pron{£p£hs£os£il£Iz} triangle has at least 
two equal sides; an \nuvo{equilateral} \pron{£b£Ikw£i£hl£at(£e)r(£e)l} triangle has three equal sides.  A \nuvo{right} 
or \nuvo{right-angled} triangle has one angle of 90\degre.\index{triangle} 
   
Pythagoras'\,\footnote{Pythagoras (6th-c \textsc{bc}) Philosopher and mathematician, born in Samos \pron{£hs£qm£os}, 
  Greece.  He settled at Crotona, Magna Graecia \pron{£bm£agn£e £hgr£Is£Q} (c. 530 \textsc{bc}) where he founded a moral 
  and religious school. He eventually fled from ther because of persecution, settling at Megapontum in Lucania 
  \pron{l£U£hk£qn£Q}. The famous theorem attributed to him was probably developed later by members of the Pythagorean 
  school, which is best known for its studies of the relations between numbers.} theorem \pron{p£p£h£f£ag£er£as 
  £h£f£Qr£em} (also known as Pythagorean \pron{p£p£b£f£ag£e£hr£Qn} theorem) states that if $\triangle ABC$ is a right 
triangle with \emph{legs} 
of lengths $a$ and $b$, and \nuvo{hypotenuse} \pron{h£p£hp£ot£enj£Uz} of length $c$, then $\dps c^2= a^2+b^2$.%   
\index{leg}\index{Pythagoras}\index{pythagorean} 
   
An \nuvo{altitude} \pron{£h£alt£itj£Ud} is a segment between a vertex and the opposite \pron{£h£op£ez£it} side or 
\nuvo{base} \pron{b£qs}, that is perpendicular to the side. It is also the length of that segment. The common point of 
the three altitudes is the \nuvo{orthocentre} \pron{£b£O£f£w£hsent£e£Z}. 
   
A \nuvo{median} \pron{£hm£Idj£en} is a line joining a vertex of a triangle to the midpoint of the opposite side. All 
three such lines coincide in the \nuvo{centroid} \pron{£hsentr£rd}. Archimedes \pron{£b£Ak£i£hm£Id£Iz} (c. 
\detantatant{287}{212} \textsc{b.c.})  obtained it as the center of gravity \pron{£hgr£av£it£i} of a triangular 
\pron{tr£p£h£a£ngj£ul£e£Z} plate of uniform density \pron{£hdens£et£i}. The centroid of the triangle is also the point 
of trisection \pron{tr£p£hsek£sn} of the three medians of the triangle. \index{Archimedes} 
   
The mid-perpendiculars of the sides of a triangle intersect at the \nuvo{circumcentre} \pron{£bs£Ek£em£hsent£e£Z} which 
is the \emph{centre} \pron{£hsent£e£Z} of the \nuvo{circumcircle} \pron{£hs£Ek£em£bs£Ekl} or \emph{circumscribed} 
\pron{£hs£Ek£em£bskr£pbd} \emph{circle}.\index{circumscribed} 
   
\subsection{Trigonometry}   
   
Trigonometry \pron{£btr£ig£e£hn£om£etr£i} is the branch of mathematics concerned with the properties of the
\emph{trigonometric} \pron{£btr£ig£en£e£hmetr£ik} \emph{ratios} \pron{£hr£q£s£i£w} or \emph{trig ratios} and their
application to the determination of the sides and angles of triangles, used in surveying \pron{s£e£hv£q£i£n}, navigation\dots%
\index{trigonometry}\index{trigonometric} \index{surveying}\index{trig ratio}
   
\dicodef{To survey} \pron{s£e£hv£q} \textbf{1} take or present a general view of. \textbf{2} examine the condition of (a 
building etc{.}), especially on behalf of a prospective buyer. \textbf{3} determine the boundaries, extent, ownership, 
etc{.}, of (a district etc{.}). 
   
The trig ratios are: \nuvo{sine} \pron{s£pn}, \nuvo{cosine} \pron{£hk£ws£pn}, \nuvo{tangent} \pron{£ht£and£j£ent}, 
cotangent \pron{£bk£w£ht£and£j£ent}, secant \pron{£hs£Ik£ent}, and cosecant \pron{£bk£w£hs£Ik£ent}. 
   
Let $ABC$ be a right triangle the hypotenuse of which is $AB$. Let §a denote $\angle BAC$. Then 
   
$$\sin§a =\dfrac{BC}{AB}=\dfrac{\text{opposite}}{\text{hypotenuse}} \qquad\mbox{}\qquad \cos§a 
=\dfrac{AC}{AB}=\dfrac{\text{adjacent}}{\text{hypotenuse}}$$ 
$$\tan§a =\dfrac{BC}{AC}=\dfrac{\text{opposite}}{\text{adjacent}}$$ 
   
There are several mnemonics \pron{n£I£hm£on£ik} (memory aids) which can help you to remember those 
formulae\,\footnote{plural of `formula' \pron{£hf£Omj£ul£e}} \pron{£hf£Omj£ul£I}. Here is an example:\index{formula}
\begin{center}   
  \textsc{s o h c a t h t o a}\\ 
  Some Officers Have Curly Auburn Hair To Offer Attraction. 
\end{center}   
\textsc{s} stands for `sine', \textsc{c} for `cosine', \textsc{t} for `tangent', \textsc{a} for `adjacent', \textsc{h} 
for `hypotenuse', and \textsc{o} for `opposite'. 
   
\subsection{Quadrilaterals}   
A \nuvo{quadrilateral} or \nuvo{trapezoid} \pron{£htr£ap£iz£rd} is any four-sided plane figure. It has four vertices. 
The \emph{diagonals} \pron{d£p£h£ag£enl} of $ABCD$ are $[AC]$ and $[BD]$.\index{diagonal}\index{trapezoid} 
   
A \nuvo{trapezium} \pron{tr£e£hp£Izj£em} is a quadrilateral with two parallel sides of unequal length. A 
\nuvo{parallelogram} \pron{£bp£ar£a£hlel£egr£am} is a quadrilateral with opposite pairs of sides parallel.  A 
\nuvo{rhombus}\,\footnote{plural: `rhombi' \pron{£hr£omb£p} or `rhombuses' \pron{£hr£omb£es£iz}.}  \pron{£hr£omb£es} is 
an equilateral parallelogram --- all its sides are equal. A \nuvo{rectangle} \pron{£hrek£bt£a£ngl} 
is a quadrilateral all the angles of which are right. A \nuvo{square} \pron{skw£P£Z} is an equilateral rectangle.%   
\footnote{In US English the British `trapezoid' is called `trapezium' and vice versa \pron{£bv£ps£i£hv£Es£e}.} 
   
If the lengths are measured with metres \pron{£hm£It£e£Z}, the \nuvo{area} \pron{£h£Pr£Q} are measured with square 
metres.  The area of a square of side $s$~m is equal to $s^2$~m$^2$ `square meter'. 
   
\subsection{Circles}   
   
A \nuvo{circle} is a closed \pron{kl£wzd} plane \nuvo{curve} \pron{k£Ev} every point of which is equidistant from a 
given fixed point, the \nuvo{centre}. In other words, a circle is the \nuvo{path} \pron{p£A£f} of a point that moves so 
as to keep a constant distance from a fixed point, the centre. 
   
Each circle comprises \pron{k£em£hpr£pz£iz} a \nuvo{radius}\,\footnote{plural `radiuses' \pron{£hr£qd£Qs£iz} or `radii' 
  \pron{£hr£qd£i£b£p}} \pron{£hr£qd£Qs} ---~the distance from any point on the circle to the centre~---, a 
\nuvo{circumference} \pron{s£e£hk£vmf(£e)r(£e)ns} ---~the boundary of the circle~---, \emph{diameters} 
\pron{d£p£h£am£it£e£Z} ---~lines crossing the circle through the centre~---, \emph{chords} \pron{k£Od} ---~lines joining 
two points on the circumference~---, and \emph{tangents} \pron{£ht£and£j£ent} ---~lines that touch the circumference at 
one point.\index{diameter}\index{chord}\index{tangent} 
   
An \nuvo{arc} \pron{£Ak} is a section of a curved line. A circle has three types of arc: a \nuvo{semicircle} 
\pron{£hsem£i£bs£Ekl}, which is exactly half of the circle; \emph{minor arcs} \pron{£hm£pn£e£Z}, which are less than the 
semicircle; and \emph{major arcs} \pron{£hm£qd£j£e£Z}, which are greater than the semicircle.%   
\index{arc!minor}\index{arc!major} 
   
The ratio of the distance all around the circle ---~the \nuvo{circumference} or \nuvo{perimeter} 
\pron{pe£hr£im£it£e£Z}~--- to the diameter is an irrational number called §p \pron{p£p}. 
   
\section{Solid geometry}   
   
Solid \pron{£hs£ol£id} geometry is the branch of geometry concerned with the properties of three-dimensional figures. 
Among these figures there are: the planes, the \emph{cubes}, the \emph{cuboids} \pron{£hkj£Ub£rd} or \emph{right 
  parallelepipeds} \pron{£hp£ar£e£blel£e£hp£pp£id}, the \emph{cylinders} \pron{£hs£il£ind£e£Z}, the \emph{pyramids} 
\pron{£hp£ir£em£id}, the \emph{spheres} \pron{sf£Q£Z}.%   
\index{cuboid}\index{cube}\index{parallelepiped}\index{cylinder}\index{pyramid}\index{sphere} 
   
A \nuvo{polyhedron} \pron{£bp£ol£i£hh£Idr£en}\,\footnote{plurals: `polyhedrons' \pron{£bp£ol£i£hh£Idr£enz} or
  `polyhedra' \pron{£bp£ol£i£hh£Idr£e}} is a solid figure.  A polyhedron is said to be \nuvo{convex} if all segments
joining any two points on its boundary lie wholly inside it.  A regular convex polyhedron or `\nuvo{platonic solid}'
\pron{pl£e£ht£on£ik} is a convex polyhedron the faces of which are regular polygons. There are only five platonic solids
\emph{viz} the regular \nuvo{tetrahedron} \pron{£btetr£e£hh£Idr£en}, the cube, the regular \nuvo{octahedron}
\pron{£b£okt£e£hh£Idr£en}, \nuvo{icosahedron} \pron{£b£pk£es£e£hh£Idr£en} and \nuvo{dodecahedron}
\pron{£bd£wd£ek£e£hh£Idr£en}.
   
The cube is a polyhedron which has eight corners ---~or `vertices'~--- twelve \emph{edges} \pron{ed£j} and six faces.\index{edge}
Each face is a square. All the edges of a cube have the same length. 
   
A cuboid is a polyhedron the faces of which are rectangles. It has eight vertices, twelve edges and six faces. If one 
wishes, one can use the more tongue-twisting words `right parallelepiped' to speak about a cuboid. 
   
A \nuvo{net} is a diagram of a hollow solid consisting of the plane shapes of the faces so arranged that the diagram 
could be folded to form the solid. 
   
\begin{fgre}
  \IG{netcubepyr}
\end{fgre}
   
\section{Coordinate system}   
   
Once a unit length is selected, we can represent points on a line by numbers. 
   
\begin{fgre}
  \IG{droitegrad}
\end{fgre}
   
A \emph{coordinate system} \pron{k£w£h£Odn£et} is any system for locating points by their coordinates with respect to 
some set of reference \pron{£hrefr£ens} points, lines, directions, 
etc.\index{system!coordinate}\index{coordinate!system} 
   
\enlargethispage{2\baselineskip} 
   
\emph{Cartesian coordinates} \pron{k£a£ht£Izj£en} or \emph{rectangular coordinates} \pron{rek£ht£a£n\-gj£u\-l£e£Z} 
provide a system for the representation of a point in plane in terms of its distance, measured along a pair of mutually 
perpendicular \emph{axes} \pron{£h£ak£Iz}, from a given \nuvo{origin} \pron{£h£or£id£j£in}. On the \emph{Cartesian 
  plane} the point \couple{a}{b} is located by measuring $a$ units along the $x$-axis \pron{£h£aks£is} and $b$ units 
along the $y$-axis, and then finding the point of intersection of the perpendiculars to the axes at those points (see 
figure below); $a$ is then the \nuvo{abscissa} \pron{£a£hbs£is£e}\,\footnote{plurals: abscissas \pron{£a£hbs£is£ez} or 
  abscissae \pron{£a£hbs£is£I}} and $b$ the \nuvo{ordinate} \pron{£h£Odn£et}. By convention, the positive directions of 
the axes point to the right and upwards, so that the four points \couple{\pm1}{\pm1} are placed as shown; by convention, 
the first \nuvo{quadrant} \pron{£hkw£odr£ent} is that in which both coordinates are positive, and the other quadrants 
are numbered anticlockwise \pron{£b£ant£i£hkl£okw£pz}\,\footnote{In a curve 
opposite in direction to the movement of the hands of a clock.} from the first.%   
\index{coordinates}\index{axis}\index{anticlockwise}\index{Cartesian}\index{rectangular}%   
\index{Cartesian!plane}\index{coordinates!Cartesian}\index{coordinates!rectangular}%   
\index{Cartesian!coordinates}%   
\index{axis!$x$-}\index{axis!$y$-}\index{axes|see{axis}} 
   
\begin{fgre}
  \IG{cartcoord}
\end{fgre}
   
\dicodef{Cartesian} of or relating to \textsc{Descartes} or his work in philosophy, science, 
and mathematics.\index{Cartesian}\index{Descartes} 
   
\Dicodef{Descartes} René \Detantatant{1596}{1650} French philosopher, mathematician, and man of science, often called 
the father of modern philosophy. In mathematics he developed the use of coordinates to locate a point in two or three 
dimensions: this enabled the techniques of algebra and calculus to be used to solve geometrical problems.  
\endinput 
   
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