% This template was created by Stephen on 27th March 2000
%
% Title:  Euclidean Geometry
% Author: Stephen Farrar
% Date:   16/4/2000
%

% Set the document class
\documentclass[12pt,a4paper]{article}

% Resize the page, to save the forests.
\setlength{\topmargin}{-1cm} \setlength{\textheight}{24cm}
\setlength{\textwidth}{15cm} \setlength{\oddsidemargin}{0.5cm}
\setlength{\evensidemargin}{0.5cm}

% Just in case we want to do Black Board Bold letters.
%\input{amssymb.sty}
\usepackage{amssymb}

% So we can do multicolumns
\usepackage{multicol}
\setlength\columnseprule{0pt}   % no vertical lines inbetween columns
\setlength\multicolsep{0.5ex}     % space preceding a multicolumn

% A command to generate the header text
\newcommand{\myheader}
{\footnotesize April 2005}

% Change the headers and footers
\pagestyle{myheadings} \markboth{\myheader}{\myheader}

% New Commands to make numbered lists easier
\newcommand{\be}{\begin{enumerate}}
\newcommand{\ee}{\end{enumerate}}

% New Commands to make in-text displayed math easier
\newcommand{\bdm}{$ \displaystyle}
\newcommand{\edm}{$}

% New Commands to do handle the black board bold sets:
\newcommand{\R}{\mathbb{R}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\Q}{\mathbb{Q}}

% New command for conjugates, arguments, etc.
\def\conj#1{\overline{#1}}
\def\arg{\,\mbox{arg}\,}
\def\Arg{\,\mbox{Arg}\,}
\def\cis{\,\mbox{cis}\,}
\def\Cis#1{\cos {#1} + i \sin {#1}}
\def\Re{\, \mbox{Re}\,}
\def\Im{\, \mbox{Im}\,}
\def\cosec{\,\mbox{cosec}\,}

% New command for the heading
\newcommand{\WorksheetHeading}[1]{{\par\noindent \bf \Large #1}}

% New command to fix spelling
\newenvironment{centre}{\begin{center}}{\end{center}}

% for f:X -> Y; the default spacing isn't great
\newcommand{\map}[2]{\,{:}\,#1\!\longrightarrow\!#2}

%--------------------------------------------------------
% Let the document begin!
\begin{document}

\begin{centre}
  \bf \Large
  Plane Geometry (J)
\end{centre}

\begin{enumerate}

\item Prove that $AC$ is perpendicular to $BD$ if and only if
$AB^{2}+CD^{2}=BC^{2}+DA^{2}$.

\item Let $D$, $E$ and $F$ be points on sides $BC$, $CA$ and $AB$
of triangle $ABC$ such that $AD$, $BE$ and $CF$ are concurrent.
Let $P$ be this point of concurrence. Prove that
$$\frac{AP}{PD}=\frac{AE}{EC}+\frac{AF}{FB}.$$

\item Let $ABC$ be a triangle and $D$, $E$, $F$ the midpoints of
$BC$, $CA$, $AB$ respectively. Prove that $\angle DAC=\angle ABE$
if and only if $\angle AFC=\angle ADB$.

\item (Quadrilateral version of Menelaus' Theorem) Let $ABCD$ be a
quadrilateral and suppose that a line $\ell$ intersects $AB$ at
$K$, $BC$ at $L$, $CD$ at $M$ and $DA$ at $N$. Prove that \[
AK\cdot BL\cdot CM\cdot DN=KB\cdot LC\cdot MD\cdot NA \]

\item (radical axis theorem - you can quote this) Three circles
pairwise intersect. Prove that the three chords common to pairs of
circles are concurrent.

\item $O$ is the midpoint of the base $BC$ of an isosceles
triangle $ABC$. A circle is drawn with centre $O$ and tangent to
the equal sides $AB$, $AC$. $P$ is a point on $AB$ and $Q$ is a
point on $AC$. If $PQ$ is also tangent to this circle, prove that
$PB\cdot CQ=(\frac{1}{2}BC)^2$.

\item Suppose the incircle of triangle $ABC$ touches sides $BC$,
$CA$ and $AB$ at points $A_1$, $B_1$ and $C_1$ respectively. $A_2$
is the reflection of $A_1$ about the perpendicular bisector of
$BC$. Similarly define $B_2$ and $C_2$. Prove that $AA_i$, $BB_i$
and $CC_i$ are concurrent for $i=1,2$.

\item (Tournament of The Towns, Junior O level, Autumn 1997) Let
$ABCD$ be a square and $X$, $Y$ be points on the sides $BC$ and
$CD$ respectively such that $AY$ bisects $\angle DAX$. Show that
$AX=BX+DY$.

\item Prove that a quadrilateral with side lengths $a$, $b$, $c$
and $d$ in that order can have a circle inscribed inside it
tangent to all four sides if and only if $a+c=b+d$.

\item Let $ABCD$ be a quadrilateral. Prove that the incircles of
$ABC$ and $CDA$ are tangent to each other if and only if the
incircles of $BCD$ and $DAB$ are tangent to each other.

\item (1998 IMO Q1) In the convex quadrilateral $ABCD$, the
diagonals $AC$ and $BD$ are perpendicular and the opposite sides
$AB$ and $DC$ are not parallel. Suppose that the point $P$, where
the perpendicular bisectors of $AB$ and $DC$ meet, is inside
$ABCD$.  Prove that $ABCD$ is a cyclic quadrilateral if and only
if the triangles $ABP$ and $CDP$ have equal areas.

\item (1998 IMO Q5) Let $I$ be the incenter of triangle $ABC$.
Let the incircle of $ABC$ touch the sides $BC$, $CA$, and $AB$ at
$K$, $L$, and $M$, respectively.  The line through $B$ parallel to
$MK$ meets the lines $LM$ and $LK$ at $R$ and $S$, respectively.
Prove that angle $RIS$ is acute.


\end{enumerate}


\end{document}
