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Construct a regular pentagon

mathmari

Well-known member
MHB Site Helper
Apr 14, 2013
4,036
Hey!! 😊

I want to construct a regular pentagon with side length $4$.

I have done the following:

We are given the segment $|AT|=4$ and we will create a point $B$ outside of $|AT|$ such that $|AB|$ is divided by $T$ in golden ratio.

We create the perpendicular line segment to $|AT|$ from $T$ with length $\overline{AT}$ and so we get the point $C$.

Then we create the midpoint $M$ of $|AT|$.

The circle around $M$ with radius $\overline{MC}$ intersects the extension of $|AT|$ from $T$ at the point $B$.

So we have that $\frac{\overline{AB}}{\overline{AT}}=\Phi$.

From that equation we get that $\overline{AB}=\overline{AT}\cdot \Phi=4\Phi$.

Then we create an isosceles triangle on the segment $|AT|$ with the two equal sides of length $\overline{AB}$ as follows:

We create a circle with center $A$ and radius $\overline{AB}$ and a circle with center $T$ and radius $\overline{AB}$. Let $P$ be the intersection point of the two circles, above of $|AB|$.
This is a triangle of type $1$.

On the sides $|AP|$ and $|TP|$ we create an isosceles triangle with two equal sides of length $4$.

Let's consider first the side $|AP|$.
We create a circle with center $A$ and radius $4$ and a circle with center $P$ and radius $4$. Let $K_1$ be the intersection point of the two circles, above of $|AP|$.
This is a triangle of type $2$.

Now we consider the side $|TP|$.
We create a circle with center $T$ and radius $4$ and a circle with center $P$ and radius $4$. Let $K_2$ be the intersection point of the two circles, above of $|TP|$.
This is a triangle of type $2$.

Now we have a regular pentagon.

Is everything correct? :unsure:

To justify that these are indeed triangles of type $1$ and $2$ respectively do we calculate the angles using the cosine rule? :unsure:
 

mathmari

Well-known member
MHB Site Helper
Apr 14, 2013
4,036
Which online program should I use to make the figure for each step?
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,739
Which online program should I use to make the figure for each step?
Hey mathmari !!

The first ones that come to my mind are:
  • TikZ, which is supported natively on this site.
  • Desmos, which is also supported natively on this site.
  • Geogebra, which I think you are already familiar with.
Following the first couple of steps, I think you have something like the following TikZ picture:
\begin{tikzpicture}
\coordinate[label=left:A] (A) at (0,0);
\coordinate[label=right:B] (B) at (6.4,0);
\coordinate[label=C] (C) at (4,4);
\coordinate[label=below:T] (T) at (4,0);
\draw[help lines] (T) rectangle +(0.4,0.4);
\draw (A) -- node[ below ]{$4$} (T) -- (B);
\draw (T) -- node[ right ]{$4$} (C);
\draw[<->, help lines] ([yshift=-1cm]A) -- node[ below ]{$4\Phi$} ([yshift=-1cm]B);
\end{tikzpicture}
Is that correct? Or am I already off? 🤔

Btw, I'm a little confused by your notation. o_O
Usually $\overline{AB}$ means the line segment between points $A$ and $B$, and not its length, which is what you seem to use it for.
And $\left|\overline{AB}\right|$ is its length, which can also be written as simply $AB$.
Writing $|AB|$ as you do, seems to imply a length as you are using absolute sign symbols, but it appears you use it for the line segment.
Where did you find that notation? :unsure:
 
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