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Euclid Book 6
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Euclid Book 6 Proposition 17a
Statement
I
f
o
n
e
s
i
d
e
(
A
B
)
o
f
a
r
e
c
t
a
n
g
l
e
(
A
B
C
D
)
i
s
t
o
t
h
e
s
i
d
e
(
E
F
)
o
f
a
s
q
u
a
r
e
(
E
F
G
H
)
a
s
t
h
e
s
i
d
e
o
f
t
h
e
s
q
u
a
r
e
i
s
t
o
t
h
e
o
t
h
e
r
s
i
d
e
(
B
C
)
o
f
t
h
e
r
e
c
t
a
n
g
l
e
(
A
B
E
F
E
F
B
C
)
,
t
h
e
n
t
h
e
s
q
u
a
r
e
a
n
d
t
h
e
r
e
c
t
a
n
g
l
e
h
a
v
e
e
q
u
a
l
a
r
e
a
s
.
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
.
}
}
,
G
e
o
m
e
t
r
i
c
A
s
s
e
r
t
i
o
n
[
{
P
o
l
y
g
o
n
[
{
A
.
,
B
.
,
C
.
,
D
.
}
]
}
,
"
E
q
u
i
a
n
g
u
l
a
r
"
]
,
G
e
o
m
e
t
r
i
c
A
s
s
e
r
t
i
o
n
[
{
P
o
l
y
g
o
n
[
{
E
.
,
F
.
,
G
.
,
H
.
}
]
}
,
"
R
e
g
u
l
a
r
"
]
,
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
[
B
.
,
C
.
]
b
.
,
E
u
c
l
i
d
e
a
n
D
i
s
t
a
n
c
e
[
E
.
,
F
.
]
c
.
,
a
.
c
.
c
.
b
.
,
A
r
e
a
[
P
o
l
y
g
o
n
[
{
A
.
,
B
.
,
C
.
,
D
.
}
]
]
A
r
e
a
[
P
o
l
y
g
o
n
[
{
E
.
,
F
.
,
G
.
,
H
.
}
]
]
,
a
.
b
.
2
c
.
A
r
e
a
[
P
o
l
y
g
o
n
[
{
A
.
,
B
.
,
C
.
,
D
.
}
]
]
A
r
e
a
[
P
o
l
y
g
o
n
[
{
E
.
,
F
.
,
G
.
,
H
.
}
]
]
Explanations
Let the three straight lines
A
,
B
,
C
be proportional, so that, as
A
is to
B
, so is
B
to
C
; I say that the rectangle contained by
A
,
C
is equal to the square on B.
Let
D
be made equal to
B
.
Then, since, as
A
is to
B
, so is
B
to
C
, and
B
is equal to
D
, therefore, as
A
is to
B
, so is
D
to
C
.
But, if four straight lines be proportional, the rectangle contained by the extremes is equal to the rectangle contained by the means.
[
V
I
.
1
6
]
Therefore the rectangle
A
,
C
is equal to the rectangle
B
,
D
.
But the rectangle
B
,
D
is the square on
B
, for
B
is equal to
D
; therefore the rectangle contained by
A
,
C
is equal to the square on
B
.
Classes
Euclid's Elements
Theorems
Geometric Algebra
EuclidBook6
Related Theorems
EuclidBook6Proposition16a
EuclidBook6Proposition16b
EuclidBook6Proposition17b
