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Nicomedes's Geometric Construction of Principal Cube Root

​
set k
2
move S to get RS = 1
1.5
verge
RS = 1.2000
AS = 1.5000
3
k
= 1.2599
This Demonstration shows Nicomedes's method for constructing a cube root using a marked ruler. (Nicomedes was born about 270 BC.) This is actually the real principal cube root. Every nonzero number also has a pair of complex conjugate cube roots.
Let
△ABC
have sides of length
k/4
, 1 and 1, where
AB=k/4
with
0<k<4
. Let
M
be the midpoint of
AB
and let
B
be the midpoint of
CD
. Let
E
be a point on the line through
C
parallel to
AB
. Let
S
be a point on the ray
BA
and
R
be the intersection of segments
DE
and
SC
. If
RS=1
, then
AS=
3
k
.
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