bibin/cphb
1
0
Fork 0
Code Issues Pull requests Releases Wiki Activity

Some fixes

Browse source
This commit is contained in:
Antti H S Laaksonen 2017-02-26 13:10:29 +02:00
parent 8a936ed246
commit 25c948597e
4 changed files with 60 additions and 45 deletions
Download patch file Download diff file Expand all files Collapse all files

14
chapter29.tex
View file

@ -14,7 +14,7 @@ and our task is to calculate its area.
For example, a possible input for the problem is as follows:
\begin{center}
\begin{tikzpicture}[scale=0.44]
\begin{tikzpicture}[scale=0.45]
\draw[fill] (6,2) circle [radius=0.1];
\draw[fill] (5,6) circle [radius=0.1];
@ -27,7 +27,7 @@ One way to approach the problem is to divide
the quadrilateral into two triangles by a straight
line between two opposite vertices:
\begin{center}
\begin{tikzpicture}[scale=0.44]
\begin{tikzpicture}[scale=0.45]
\draw[fill] (6,2) circle [radius=0.1];
\draw[fill] (5,6) circle [radius=0.1];
@ -41,8 +41,8 @@ line between two opposite vertices:
After this, it suffices to sum the areas
of the triangles.
The area of a triangle can be calculated,
for example, using \key{Heron's formula}\footnote{Heron of Alexandria
(c. 10--70) was a Greek mathematician.}
for example, using \key{Heron's formula}
%\footnote{Heron of Alexandria (c. 10--70) was a Greek mathematician.}
\[ \sqrt{s (s-a) (s-b) (s-c)},\]
where $a$, $b$ and $c$ are the lengths
of the triangle's sides and
@ -57,7 +57,7 @@ two arbitrary opposite vertices.
For example, in the following situation,
the division line is outside the quadrilateral:
\begin{center}
\begin{tikzpicture}[scale=0.44]
\begin{tikzpicture}[scale=0.45]
\draw[fill] (6,2) circle [radius=0.1];
\draw[fill] (3,2) circle [radius=0.1];
@ -70,7 +70,7 @@ the division line is outside the quadrilateral:
\end{center}
However, another way to draw the line works:
\begin{center}
\begin{tikzpicture}[scale=0.44]
\begin{tikzpicture}[scale=0.45]
\draw[fill] (6,2) circle [radius=0.1];
\draw[fill] (3,2) circle [radius=0.1];
@ -573,7 +573,7 @@ along the boundary of the polygon.
\index{Pick's theorem}
\key{Pick's theorem} \cite{pic99} provides another way to calculate
\key{Pick's theorem} provides another way to calculate
the area of a polygon provided that all vertices
of the polygon have integer coordinates.
According to Pick's theorem, the area of the polygon is