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Euclid Book 6
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Euclid Book 6 Proposition 26
Statement
I
f
t
w
o
s
i
m
i
l
a
r
p
a
r
a
l
l
e
l
o
g
r
a
m
s
(
A
B
C
D
,
A
E
F
G
)
h
a
v
e
a
v
e
r
t
e
x
(
A
)
a
n
d
s
i
d
e
s
(
A
B
,
A
E
;
A
D
,
A
G
)
a
d
j
a
c
e
n
t
t
o
t
h
a
t
v
e
r
t
e
x
i
n
c
o
m
m
o
n
,
t
h
e
n
t
h
e
y
l
i
e
o
n
t
h
e
s
a
m
e
d
i
a
g
o
n
a
l
(
A
C
)
.
Computational Explanation
G
e
o
m
e
t
r
i
c
S
c
e
n
e
[
{
{
A
.
,
B
.
,
C
.
,
D
.
,
E
.
,
F
.
,
G
.
}
,
{
}
}
,
{
L
i
n
e
[
{
{
A
.
,
E
.
,
B
.
}
,
{
A
.
,
G
.
,
D
.
}
}
]
,
G
e
o
m
e
t
r
i
c
A
s
s
e
r
t
i
o
n
[
{
P
a
r
a
l
l
e
l
o
g
r
a
m
[
{
A
.
,
B
.
,
C
.
,
D
.
}
]
,
P
a
r
a
l
l
e
l
o
g
r
a
m
[
{
A
.
,
E
.
,
F
.
,
G
.
}
]
}
,
"
S
i
m
i
l
a
r
"
]
,
L
i
n
e
[
{
A
.
,
C
.
}
]
}
,
{
F
.
∈
L
i
n
e
[
{
A
.
,
C
.
}
]
}
]
F
.
∈
L
i
n
e
[
{
A
.
,
C
.
}
]
Explanations
For suppose it is not, but, if possible, let
A
H
C
be the diameter of
A
B
C
D
, let
G
F
be produced and carried through to
H
, and let
H
K
be drawn through
H
parallel to either of the straight lines
A
D
,
B
C
.
[
I
.
3
1
]
Since, then,
A
B
C
D
is about the same diameter with
K
G
, therefore, as
D
A
is to
A
B
, so is
G
A
to
A
K
.
[
V
I
.
2
4
]
But also, because of the similarity of
A
B
C
D
,
E
G
, as
D
A
is to
A
B
, so is
G
A
to
A
E
; therefore also, as
G
A
is to
A
K
, so is
G
A
to
A
E
.
[
V
.
1
1
]
Therefore
G
A
has the same ratio to each of the straight lines
A
K
,
A
E
.
Therefore
A
E
is equal to
A
K
[
V
.
9
]
, the less to the greater: which is impossible.
Therefore
A
B
C
D
cannot but be about the same diameter with
A
F
; therefore the parallelogram
A
B
C
D
is about the same diameter with the parallelogram
A
F
.
Classes
Euclid's Elements
Theorems
Quadrilaterals
EuclidBook6
