B divides AC into the golden ratio if

1
B divides AC into the golden ratio if
2
Are these definitions equivalent?
3
This definition screams AREA! (Area of the square
equals area of the rectangle)
4
Canonical construction of the golden ratio is
proven in Euclid (Construct a square, bisect the
base, construct semidiagonal, swing this length
to base)
5
Euclid starts with the following lemma. Extend a
line segment AB out to a point C, then find the
midpoint of AB. (C is an arbitrary distance from
B). Then
MC2MB2ACBC
One website says it must have taken a lot of
brain power to prove this. But again, think area
6
MC2MB2ACBC
Here weve drawn the square on MC, and the
rectangle on AC and BC. Lets draw a vertical
line through B
7
1
4
2
3
5
MC2MB2ACBC
MC2 is the sum of 1234 MB2 is area 1 The
rectangle is 235234 Now Euclid proves the
golden ratio construction
8
MC2MB2ACBC (lemma)

MC2MD2MB2BD2MB2AB2

MB2AB2MB2ACBC
AB2ACBC
The lemma is also used in the next proof
9
B
C
D
A
AD?ACAB2
10
That last proof is used to prove a theorem in
solid geometry. Neither the lemma or golden
ratio construction is used again. Do you suppose
someone trying to prove that solid geometry
theorem realized he needed that third theorem,
then realized he needed the lemma? And maybe
later, he or someone else realized the lemma
would be useful in proving the golden ratio
construction. Its also interesting that Euclid
does not use this construction to construct a
regular pentagon, even though constructing a
regular pentagon automatically involves
constructing the golden ratio.
11
Another construction of the golden ratio
Draw a circle, then using the same radius on the
compass, strike off six equal arcs. If we
connected consecutive arcs, wed get a regular
hexagon, but were only going to connect every
other arc.
12
This gives a regular triangle. Now connect the
center of the circle with two of the unused tic
marks. This is only to find the midpoint of two
sides of the triangle.
13
Now draw a line through those two midpoints and
extend it to the circle on one side
14
Then B divides AC into the golden ratio
15
This construction was discovered by George Odom
around 1979. George was an amateur mathematician
and also a sculptor. He spent quite a bit of his
life in an institution for the mentally
insane(!). He was acquainted with H.S.M.
Coxeter, the premier geometer of the 20th
century. Coxeter had not seen this proof, so
submitted it in Odoms name to the Mathematical
Association of America (in Odoms name), and it
was published in their monthly in 1980. Here are
a couple ways to prove that it works
16
Scale figure so BD2. Then MB1, MD
Now find OD (which OC), OM, MC, then AC
A second proof follows
17
Interior angles cutting off same arc
http//www.cut-the-knot.org/do_you_know/GoldenRati
o.shtml
18
Proof Without Words http//www.cut-the-knot.org/do
_you_know/GoldenRatio.shtml
19
Heres another construction. Draw a vertical
line segment
20
Draw a circle centered at each end with radius
equal to the length of the line segment
21
Extend the line segment to intersect the upper
circle
22
Draw a circle with center at the center of the
lower circle and passing through that upper point
of the line segment
23
Draw a line through the intersection of the two
smaller circles and extend it to the larger
circle
24
To see why this works, draw DA and DB to
intersect the outer circle
D
B
A
O
Now angle ODB is inscribed in a semicircle, so is
a right angle. Also OD is twice OB, so angle ODB
is 30o so the triangle is equilateral, and this
is equivalent to Odoms construction.
25
Incidentally, the fact that an angle inscribed in
a semicircle is a right angle is attributed to
Thales, and is one of the first theorems ever
proved in Greek geometry. Ive often wondered
what axioms and postulates Thales used, and what
logical constructs he was familiar with
(Aristotelian logic was not formalized by
Aristotle until over a hundred years later), but
there are no records to answer that
question. One additional construction of the
golden ratio is the rusty compass construction.
A rusty compass is one that has a frozen
hinge. It will still draw circles, but only of
one radius. Constructions of this type were
studied by Persian mathematicians in the 12th
century.
26
It is possible to do everything with a rusty
compass and straightedge that can be done with a
standard compass and straightedge except draw
circles of a given radius. The following
construction is from a neat website called
Cut-the-knot. I copied their construction, but
it did not reproduce very well.
27
RUSTY COMPASS CONSTRUCTION OF GOLDEN RATIO
http//www.cut-the-knot.org/do_you_know/GoldenRati
o.shtml
28

• http//web.aurora.edu/bdillon/golden.ppt
• http//web.aurora.edu/bdillon/pentagon.ppt
• http//web.aurora.edu/bdillon/math.htm