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In Robert Dixon's Mathographics, a regular pentagon is constructed with straightedge and compass only. It is the pentagon $ABCDE$ pictured below.

I am having trouble seeing why the central angles are all $72^\circ$, though. Can anyone provide the proof?

Also, does anyone know who this construction is due to? I haven't seen it anywhere, other than in Dixon's book; is it Dixon's result?

enter image description here

The result appears much more impressive without all the labels (which are slightly misplaced, please excuse this); however, I provided those so that answering would be easier. Also, it makes it easy to describe the steps in the construction:

1) Draw a circle (the red one) with center $h$.

2) Draw the perpendicular lines $\ell_1$ and $\ell_2$ through $h$. Locate the points of intersection $f$, $B$, and $g$ with the red circle.

3) Bisect the line segment $gh$. Denote the center by $a$.

4) Draw the green circle with center $a$ and radius $ah$.

5) Draw the other green circle (as in (3) and 4)).

6) Draw the line segment through $f$ and $a$.

7) Locate the points of intersection $b$ and $c$ of the line segment with the circle constructed in step 4).

8) Draw the blue arcs (both have center at $f$ and the radii are $fb$ and $fc$).

9) Locate the points of intersection $A$, $C$, $D$, and $E$.


I actually have the solution to the first question, and will post it unless a more elegant explanation is provided (which is probably likely). However, I find this construction particularly beautiful, and would like to know who it is attributed to (Dixon doesn't say, explicitly).

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    Would you be alright with a coordinate geometry proof?2012-01-01
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    @J.M. Yes, that would be interesting.2012-01-01
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    The _idea_ of the construction seems to me to be that the side (Ef) of a regular _decagon_ has a certain nice(ish) relation to the corner radius of the decagon. This side is then being constructed using the Pythagorean triangle ahf.2012-01-01
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    On MathOpenRef there's [an animated and detailed version of this construction](http://www.mathopenref.com/constinpentagon.html). Unfortunately, they don't provide any references or justifications. On cut the knot there's [a closely related construction](http://www.cut-the-knot.org/pythagoras/PentagonConstruction.shtml) attributed to Y. Hirano (19th century).2012-01-02
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    I admire this question, but I had great difficulty seeing the colors you mention. I therefore reproduced your diagram as closely as I could and emphasized the colors. I hope you don't mind, but if you do mind go ahead and put your own graphic back. +12015-07-05
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    @RoryDaulton It looks good, thanks!2015-07-05

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