Math & Science ⇒ theorem and proof

[abu aasiyah]

Hello,

Could someone help me with setting up a theorem and proof and make it look nice. I have that following code:

Code: Select all

Code, edit and compile here:

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\documentclass[12pt,a4paper]{article} 
\usepackage[latin1]{inputenc} 
\usepackage[danish]{babel} % danske overskrifter
\usepackage[T1]{fontenc} 
\usepackage{lmodern} 
\usepackage{graphicx} 
\usepackage{mathtools}
\usepackage{amsmath}
\usepackage{amsfonts}
\newtheorem{startthe}{Sætning}
\newtheorem{cor}{Korollar}
\newtheorem{defn}{Definition}
\usepackage{ulem}
\usepackage{textcomp}
\usepackage{tikz}
\usepackage{color}
\usepackage{arcs}
\newenvironment{proof}[1][Bevis 1]{\begin{trivlist}
\item[\hskip \labelsep {\bfseries #1}]}{\end{trivlist}}
\begin{startthe}
Idet $k_{n}$ betegner sidelængden i den omskrevne $n$-kant i enhedsvirklen, så gælder at:
$k_{2n}=\sqrt{2-\sqrt{4-k_{n}^{2}}}$
\end{startthe}
\begin{proof}
Lad os tage udgangspunkt i figur 2. Det ses, at $\angle A$ er ret, fordi den spænder over buen $EAD$. Projiceres punkt $A$ ned på $|AE|$ ses, at $\angle D$ også bliver ret. Idet $\angle B$ indgår i $\triangle ADB$ og $\triangle BAE$ så kan disse to treknater siges at være ensvinklede. Dette betyder, at vi kan multiplicere den ene trekants sidelængder med en faktor og få den anden trekants sidelængder, dvs:
\begin{alignat*}{2}
\frac{AB}{BD}&=\frac{BE}{AB} \qquad &\Leftrightarrow \\
AB^{2}&=BE\cdot BD \qquad &\Leftrightarrow \\
AB&=\sqrt{BE\cdot BD} \qquad &\Leftrightarrow \\
k_{2n}&=\sqrt{2\cdot BD}
\end{alignat*}
\noindent Idet det ses, at $BD=1-OD$, kan vi anvende pythagoras' sætning på $\triangle DOA$ og finde et udtryk for $BD$. Vi får:
\begin{align*}
BD=1-\sqrt{OA^{2}-AD^{2}}=1-\sqrt{1-\left( \frac{1}{2}k_{n} \right)^{2}}
\end{align*}
\noindent Indsættes denne værdi i udtrykkket for $k_{2n}$ fås:
 
 
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Run LaTeX here

There are 4 things which I would like to change:

1)change the italic script to normal in "sætning"
2)make a grey box around the theorem, which has a length equal to the ordinary text appearing in the document
3)Make the font of "sætning" (theorem) and "bevis" (proof) the same
4)Have the equation in "Bevis" appear in a new line and centered but not too far down

Sorry for all the points but I hope someone can assist me.

Thanks

[daleif]

use ntheorem, and define the proof env in a much better way. Both are explained in my Danish LaTeX book, just do a google for latexbog plus my name