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Difference between revisions of "2002 IMO Problems/Problem 2"

(→‎Solution)
(→‎Solution)
Line 7: Line 7:
 
:<math>\text{By construction, AEOF is a rhombus with } 60^\circ - 120^\circ \text{angles}</math>
 
:<math>\text{By construction, AEOF is a rhombus with } 60^\circ - 120^\circ \text{angles}</math>
 
:<math>\text{ Consequently, we may set } s = AO = AE = AF = EO = EF</math>
 
:<math>\text{ Consequently, we may set } s = AO = AE = AF = EO = EF</math>
 +
:<math>\documentclass{article}
 +
\usepackage[pdftex]{graphicx}
 +
\usepackage{asymptote}
 +
\begin{document}
 +
Hello. 
 +
I like to make pics with Asymptote like this one:
 +
\begin{figure}[h]
 +
  \begin{asy}
 +
    import graph;
 +
    size(1inch);
 +
    filldraw(circle((0,0),1),yellow,black);
 +
    fill(circle((-.3,.4),.1),black);
 +
    fill(circle((.3,.4),.1),black);
 +
    draw(arc((0,0),.5,-140,-40));
 +
  \end{asy}
 +
\end{figure}
 +
\par It makes me happy,
 +
since I can still type my normal LaTeX stuff around it:
 +
\(\int_0^{\pi}{\sin{x}}\,dx=2\)
 +
\end{document}\documentclass{article}
 +
\usepackage[pdftex]{graphicx}
 +
\usepackage{asymptote}
 +
\begin{document}
 +
Hello. 
 +
I like to make pics with Asymptote like this one:
 +
\begin{figure}[h]
 +
  \begin{asy}
 +
    import graph;
 +
    size(1inch);
 +
    filldraw(circle((0,0),1),yellow,black);
 +
    fill(circle((-.3,.4),.1),black);
 +
    fill(circle((.3,.4),.1),black);
 +
    draw(arc((0,0),.5,-140,-40));
 +
  \end{asy}
 +
\end{figure}
 +
\par It makes me happy,
 +
since I can still type my normal LaTeX stuff around it:
 +
\(\int_0^{\pi}{\sin{x}}\,dx=2\)
 +
\end{document}</math>

Revision as of 11:13, 7 October 2022

Problem

$\text{BC is a diameter of a circle center O. A is any point on the circle with } \angle AOC \not\le 60^\circ$
$\text{EF is the chord which is the perpendicular bisector of AO. D is the midpoint of the minor arc AB. The line through}$
$\text{O parallel to AD meets AC at J. Show that J is the incenter of triangle CEF.}$

Solution

$\text{By construction, AEOF is a rhombus with } 60^\circ - 120^\circ \text{angles}$
$\text{ Consequently, we may set } s = AO = AE = AF = EO = EF$
$\documentclass{article}

\usepackage[pdftex]{graphicx} \usepackage{asymptote} \begin{document} Hello. I like to make pics with Asymptote like this one: \begin{figure}[h] 

 \begin{asy}
   import graph;
   size(1inch);
   filldraw(circle((0,0),1),yellow,black);
   fill(circle((-.3,.4),.1),black);
   fill(circle((.3,.4),.1),black);
   draw(arc((0,0),.5,-140,-40));
 \end{asy}

\end{figure} \par It makes me happy, since I can still type my normal LaTeX stuff around it: \(\int_0^{\pi}{\sin{x}}\,dx=2\) \end{document}\documentclass{article} \usepackage[pdftex]{graphicx} \usepackage{asymptote} \begin{document} Hello. I like to make pics with Asymptote like this one: \begin{figure}[h] 

 \begin{asy}
   import graph;
   size(1inch);
   filldraw(circle((0,0),1),yellow,black);
   fill(circle((-.3,.4),.1),black);
   fill(circle((.3,.4),.1),black);
   draw(arc((0,0),.5,-140,-40));
 \end{asy}

\end{figure} \par It makes me happy, since I can still type my normal LaTeX stuff around it: \(\int_0^{\pi}{\sin{x}}\,dx=2\) \end{document}$ (Error compiling LaTeX. Unknown error_msg)

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