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A collection of classical geometry in computable formats along with code and diagrams.
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Euclid Book 5
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Euclid Book 5 Proposition 14
Statement
I
f
a
f
i
r
s
t
m
a
g
n
i
t
u
d
e
h
a
s
t
o
a
s
e
c
o
n
d
t
h
e
s
a
m
e
r
a
t
i
o
a
s
a
t
h
i
r
d
h
a
s
t
o
a
f
o
u
r
t
h
(
A
B
C
D
E
F
G
H
)
,
a
n
d
t
h
e
f
i
r
s
t
i
s
g
r
e
a
t
e
r
t
h
a
n
o
r
e
q
u
a
l
t
o
t
h
e
t
h
i
r
d
(
A
B
≥
E
F
)
,
t
h
e
n
t
h
e
s
e
c
o
n
d
i
s
a
l
s
o
g
r
e
a
t
e
r
t
h
a
n
o
r
e
q
u
a
l
t
o
t
h
e
f
o
u
r
t
h
(
C
D
≥
G
H
)
.
Computational Explanation
G
e
o
m
e
t
r
i
c
S
c
e
n
e
{
{
A
.
,
B
.
,
C
.
,
D
.
,
E
.
,
F
.
,
G
.
,
H
.
}
,
{
a
.
,
b
.
,
c
.
,
d
.
}
}
,
S
t
y
l
e
[
L
i
n
e
[
{
A
.
,
B
.
}
]
,
R
e
d
]
,
S
t
y
l
e
[
L
i
n
e
[
{
C
.
,
D
.
}
]
,
R
e
d
]
,
S
t
y
l
e
[
L
i
n
e
[
{
E
.
,
F
.
}
]
,
B
l
u
e
]
,
S
t
y
l
e
[
L
i
n
e
[
{
G
.
,
H
.
}
]
,
B
l
u
e
]
,
E
u
c
l
i
d
e
a
n
D
i
s
t
a
n
c
e
[
A
.
,
B
.
]
a
.
,
E
u
c
l
i
d
e
a
n
D
i
s
t
a
n
c
e
[
C
.
,
D
.
]
b
.
,
E
u
c
l
i
d
e
a
n
D
i
s
t
a
n
c
e
[
E
.
,
F
.
]
c
.
,
E
u
c
l
i
d
e
a
n
D
i
s
t
a
n
c
e
[
G
.
,
H
.
]
d
.
,
a
.
b
.
c
.
d
.
,
a
.
≥
c
.
,
{
E
u
c
l
i
d
e
a
n
D
i
s
t
a
n
c
e
[
C
.
,
D
.
]
≥
E
u
c
l
i
d
e
a
n
D
i
s
t
a
n
c
e
[
G
.
,
H
.
]
,
b
.
≥
d
.
}
C
o
n
c
l
u
s
i
o
n
E
u
c
l
i
d
e
a
n
D
i
s
t
a
n
c
e
[
C
.
,
D
.
]
≥
E
u
c
l
i
d
e
a
n
D
i
s
t
a
n
c
e
[
G
.
,
H
.
]
Explanations
For let a first magnitude
A
have the same ratio to a second
B
as a third
C
has to a fourth
D
; and let
A
be greater than
C
; I say that
B
is also greater than
D
.
For, since
A
is greater than
C
, and
B
is another, chance, magnitude, therefore
A
has to
B
a greater ratio than
C
has to
B
.
[
V
.
8
]
But, as
A
is to
B
, so is
C
to
D
; therefore
C
has also to
D
a greater ratio than
C
has to
B
.
[
V
.
1
3
]
But that to which the same has a greater ratio is less;
[
V
.
1
0
]
therefore
D
is less than
B
; so that
B
is greater than
D
.
Similarly we can prove that, if
A
be equal to
C
,
B
will also be equal to
D
; and, if
A
be less than
C
,
B
will also be less than
D
.
Classes
Euclid's Elements
Theorems
Geometric Algebra
EuclidBook5
