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Euclid Book 6
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Euclid Book 6 Proposition 14b
Statement
T
w
o
p
a
r
a
l
l
e
l
o
g
r
a
m
s
(
A
B
C
D
,
E
F
G
H
)
w
i
t
h
a
p
a
i
r
o
f
e
q
u
a
l
a
n
g
l
e
s
(
∠
B
A
D
∠
F
E
H
)
,
h
a
v
i
n
g
t
h
e
s
i
d
e
s
a
d
j
a
c
e
n
t
t
o
t
h
e
e
q
u
a
l
a
n
g
l
e
s
i
n
v
e
r
s
e
l
y
p
r
o
p
o
r
t
i
o
n
a
l
(
A
B
E
F
E
H
A
D
)
,
a
r
e
e
q
u
a
l
i
n
a
r
e
a
.
Computational Explanation
G
e
o
m
e
t
r
i
c
S
c
e
n
e
{
{
A
.
,
B
.
,
C
.
,
D
.
,
E
.
,
F
.
,
G
.
,
H
.
}
,
{
}
}
,
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
[
{
E
.
,
F
.
,
G
.
,
H
.
}
]
,
P
l
a
n
a
r
A
n
g
l
e
[
{
B
.
,
A
.
,
D
.
}
]
P
l
a
n
a
r
A
n
g
l
e
[
{
F
.
,
E
.
,
H
.
}
]
,
E
u
c
l
i
d
e
a
n
D
i
s
t
a
n
c
e
[
A
.
,
B
.
]
E
u
c
l
i
d
e
a
n
D
i
s
t
a
n
c
e
[
E
.
,
F
.
]
E
u
c
l
i
d
e
a
n
D
i
s
t
a
n
c
e
[
E
.
,
H
.
]
E
u
c
l
i
d
e
a
n
D
i
s
t
a
n
c
e
[
A
.
,
D
.
]
,
{
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
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
A
B
,
B
C
be equal and equiangular parallelograms having the angles at
B
equal, and let
D
B
,
B
E
be placed in a straight line; therefore
F
B
,
B
G
are also in a straight line.
[
I
.
1
4
]
Let
G
B
be to
B
F
as
D
B
to
B
E
; I say that the parallelogram
A
B
is equal to the parallelogram
B
C
.
For since, as
D
B
is to
B
E
, so is
G
B
to
B
F
, while, as
D
B
is to
B
E
, so is the parallelogram
A
B
to the parallelogram
F
E
,
[
V
I
.
1
]
and, as
G
B
is to
B
F
, so is the parallelogram
B
C
to the parallelogram
F
E
,
[
V
I
.
1
]
therefore also, as
A
B
is to
F
E
, so is
B
C
to
F
E
;
[
V
.
1
1
]
therefore the parallelogram
A
B
is equal to the parallelogram
B
C
.
[
V
.
9
]
Classes
Euclid's Elements
Theorems
Quadrilaterals
EuclidBook6
Related Theorems
EuclidBook6Proposition14a
EuclidBook6Proposition15b
