Question 830175
<pre>
{{{drawing(400,400,-1.5,1.5,-1.5,1.5,

line(0,1,-0.95105652,0.30901699),

locate(-.05,1.15,A), 

line(-0.95105652,0.30901699,-0.58778525,-0.80901699),

locate(-1.04,.37,B),

line(-0.58778525,-0.80901699,0.58778525,-0.80901699),

locate(-.62,-.82,C),

line(0.58778525,-0.80901699,0.95105652,0.30901699),


locate(.6,-.82,D),  locate(1,.37,E),

line(0.95105652,0.30901699,0,1), 

green(line(0,1,-0.58778525,-0.80901699), line(0,1,0.58778525,-0.80901699))



)}}}

I'll just tell you how.  You can write up a two-column proof.

All sides of a regular polygon are congruent.  All interior angles of
a regular polynomial are congruent.

so AB &#8773; AE, BC &#8773; ED, &#8736;B &#8773; &#8736;E, therefore &#916;ABC &#8773; &#916;AED by SAS, 
so AC &#8773; AD by cpct.

Draw in diagonal CE.

{{{drawing(400,400,-1.5,1.5,-1.5,1.5,

line(0,1,-0.95105652,0.30901699),

locate(-.05,1.15,A), 

line(-0.95105652,0.30901699,-0.58778525,-0.80901699),

locate(-1.04,.37,B),

line(-0.58778525,-0.80901699,0.58778525,-0.80901699),

locate(-.62,-.82,C),

line(0.58778525,-0.80901699,0.95105652,0.30901699),


locate(.6,-.82,D),  locate(1,.37,E),

line(0.95105652,0.30901699,0,1), 

green(line(0,1,-0.58778525,-0.80901699), line(0,1,0.58778525,-0.80901699)),

red(line(-0.58778525,-0.80901699,0.95105652,0.30901699))

)}}}

Now &#916;CDE &#8773; &#916;ABC by SAS and AC &#8773; CE

Draw in BE

{{{drawing(400,400,-1.5,1.5,-1.5,1.5,

line(0,1,-0.95105652,0.30901699),

locate(-.05,1.15,A), 

line(-0.95105652,0.30901699,-0.58778525,-0.80901699),

locate(-1.04,.37,B),

line(-0.58778525,-0.80901699,0.58778525,-0.80901699),

locate(-.62,-.82,C),

line(0.58778525,-0.80901699,0.95105652,0.30901699),


locate(.6,-.82,D),  locate(1,.37,E),

line(0.95105652,0.30901699,0,1), 

green(line(0,1,-0.58778525,-0.80901699), line(0,1,0.58778525,-0.80901699)),

red(line(-0.58778525,-0.80901699,0.95105652,0.30901699),
line(-0.95105652,0.30901699,0.95105652,0.30901699))

)}}}

Prove &#916;CDE &#8773; &#916;ABE the same way and then BE &#8773; CE

Finally draw in BD

{{{drawing(400,400,-1.5,1.5,-1.5,1.5,

line(0,1,-0.95105652,0.30901699),

locate(-.05,1.15,A), 

line(-0.95105652,0.30901699,-0.58778525,-0.80901699),

locate(-1.04,.37,B),

line(-0.58778525,-0.80901699,0.58778525,-0.80901699),

locate(-.62,-.82,C),

line(0.58778525,-0.80901699,0.95105652,0.30901699),


locate(.6,-.82,D),  locate(1,.37,E),

line(0.95105652,0.30901699,0,1), 

green(line(0,1,-0.58778525,-0.80901699), line(0,1,0.58778525,-0.80901699)),

red(line(-0.58778525,-0.80901699,0.95105652,0.30901699),
line(-0.95105652,0.30901699,0.95105652,0.30901699)),
blue(line(-0.95105652,0.30901699,0.58778525,-0.80901699))



)}}}

And now &#916;BCD &#8773; &#916;ABE and so BE &#8773; BD.  Now write it up as a 
two-column proof.

----------------------------------------

Are any two diagonals congruent in any regular polygon?

Take a look at a regular nonagon (nine sided regular polygon):

{{{drawing(400,400,-1.5,1.5,-1.5,1.5,

line(0,1,-0.64278761,0.76604444),
line(-0.64278761,0.76604444,-0.98480775,0.17364818),
line(-0.98480775,0.17364818,-0.8660254,-0.5),
line(-0.8660254,-0.5,-0.34202014,-0.93969262),
line(-0.34202014,-0.93969262,0.34202014,-0.93969262),
line(0.34202014,-0.93969262,0.8660254,-0.5),
line(0.8660254,-0.5,0.98480775,0.17364818),
line(0.98480775,0.17364818,0.64278761,0.76604444),
line(0.64278761,0.76604444,0,1),
green(line(-0.8660254,-0.5,0.34202014,-0.93969262),
line(-0.8660254,-0.5,0.64278761,0.76604444))

)}}}

No way those two diagonals are congruent!  The only
regular polygons that have all diagonals congruent are
regular polygons with sides 4 and 5.  (A 4-sided
regular polygon is a square).

Edwin</pre>