1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
|
```
% pandoc -f latex -t native
\newtheorem{theorem}{Theorem}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\theoremstyle{remark}
\newtheorem{remark}{Remark}
\begin{definition}[right-angled triangles] \label{def:tri}
A \emph{right-angled triangle} is a triangle whose sides of length~\(a\), \(b\) and~\(c\), in some permutation of order, satisfies \(a^2+b^2=c^2\).
\end{definition}
\begin{lemma}
The triangle with sides of length~\(3\), \(4\) and~\(5\) is right-angled.
\end{lemma}
\begin{proof}
This lemma follows from \cref{def:tri} since \(3^2+4^2=9+16=25=5^2\).
\end{proof}
\begin{theorem}[Pythagorean triplets] \label{thm:py}
Triangles with sides of length \(a=p^2-q^2\), \(b=2pq\) and \(c=p^2+q^2\) are right-angled triangles.
\end{theorem}
\begin{remark}
These are all pretty interesting facts.
\end{remark}
^D
[Div ("def:tri",["definition"],[])
[Para [Strong [Str "Definition",Space,Str "1"],Space,Str "(right-angled",Space,Str "triangles).",Space,Space,Str "A",Space,Emph [Str "right-angled",Space,Str "triangle"],Space,Str "is",Space,Str "a",Space,Str "triangle",Space,Str "whose",Space,Str "sides",Space,Str "of",Space,Str "length\160",Math InlineMath "a",Str ",",Space,Math InlineMath "b",Space,Str "and\160",Math InlineMath "c",Str ",",Space,Str "in",Space,Str "some",Space,Str "permutation",Space,Str "of",Space,Str "order,",Space,Str "satisfies",Space,Math InlineMath "a^2+b^2=c^2",Str "."]]
,Div ("",["lemma"],[])
[Para [Strong [Str "Lemma",Space,Str "2"],Str ".",Space,Space,Emph [Str "The",Space,Str "triangle",Space,Str "with",Space,Str "sides",Space,Str "of",Space,Str "length\160",Math InlineMath "3",Str ",",Space,Math InlineMath "4",Space,Str "and\160",Math InlineMath "5",Space,Str "is",Space,Str "right-angled."]]]
,Div ("",["proof"],[])
[Para [Emph [Str "Proof."],Space,Str "This",Space,Str "lemma",Space,Str "follows",Space,Str "from",Space,Link ("",[],[("reference-type","ref"),("reference","def:tri")]) [Str "Definition\160\&1"] ("#def:tri",""),Space,Str "since",Space,Math InlineMath "3^2+4^2=9+16=25=5^2",Str ".",Str "\160\9723"]]
,Div ("thm:py",["theorem"],[])
[Para [Strong [Str "Theorem",Space,Str "3"],Space,Str "(Pythagorean",Space,Str "triplets).",Space,Space,Emph [Str "Triangles",Space,Str "with",Space,Str "sides",Space,Str "of",Space,Str "length",Space,Math InlineMath "a=p^2-q^2",Str ",",Space,Math InlineMath "b=2pq",Space,Str "and",Space,Math InlineMath "c=p^2+q^2",Space,Str "are",Space,Str "right-angled",Space,Str "triangles."]]]
,Div ("",["remark"],[])
[Para [Emph [Str "Remark",Space,Str "1"],Str ".",Space,Space,Str "These",Space,Str "are",Space,Str "all",Space,Str "pretty",Space,Str "interesting",Space,Str "facts."]]]
```
|
