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Euclid Book 6
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Euclid Book 6 Proposition 2b
Statement
I
f
t
w
o
s
i
d
e
s
(
A
B
,
A
C
)
o
f
a
t
r
i
a
n
g
l
e
(
△
A
B
C
)
a
r
e
c
u
t
p
r
o
p
o
r
t
i
o
n
a
l
l
y
,
t
h
e
n
t
h
e
l
i
n
e
(
D
E
)
j
o
i
n
i
n
g
t
h
e
p
o
i
n
t
s
o
f
i
n
t
e
r
s
e
c
t
i
o
n
i
s
p
a
r
a
l
l
e
l
t
o
t
h
e
t
h
i
r
d
s
i
d
e
(
B
C
)
.
Computational Explanation
G
e
o
m
e
t
r
i
c
S
c
e
n
e
{
{
A
.
,
B
.
,
C
.
,
D
.
,
E
.
}
,
{
}
}
,
T
r
i
a
n
g
l
e
[
{
A
.
,
B
.
,
C
.
}
]
,
L
i
n
e
[
{
{
A
.
,
D
.
,
B
.
}
,
{
A
.
,
E
.
,
C
.
}
}
]
,
L
i
n
e
[
{
D
.
,
E
.
}
]
,
E
u
c
l
i
d
e
a
n
D
i
s
t
a
n
c
e
[
A
.
,
D
.
]
E
u
c
l
i
d
e
a
n
D
i
s
t
a
n
c
e
[
D
.
,
B
.
]
E
u
c
l
i
d
e
a
n
D
i
s
t
a
n
c
e
[
A
.
,
E
.
]
E
u
c
l
i
d
e
a
n
D
i
s
t
a
n
c
e
[
E
.
,
C
.
]
,
{
G
e
o
m
e
t
r
i
c
A
s
s
e
r
t
i
o
n
[
{
L
i
n
e
[
{
D
.
,
E
.
}
]
,
L
i
n
e
[
{
B
.
,
C
.
}
]
}
,
"
P
a
r
a
l
l
e
l
"
]
}
G
e
o
m
e
t
r
i
c
A
s
s
e
r
t
i
o
n
[
{
L
i
n
e
[
{
D
.
,
E
.
}
]
,
L
i
n
e
[
{
B
.
,
C
.
}
]
}
,
P
a
r
a
l
l
e
l
]
Explanations
Let the sides
A
B
,
A
C
of the triangle
A
B
C
be cut proportionally, so that, as
B
D
is to
D
A
, so is
C
E
to
E
A
; and let
D
E
be joined.
I say that
D
E
is parallel to
B
C
. For, with the same construction, since, as
B
D
is to
D
A
, so is
C
E
to
E
A
, but, as
B
D
is to
D
A
, so is the triangle
B
D
E
to the triangle
A
D
E
, and, as
C
E
is to
E
A
, so is the triangle
C
D
E
to the triangle
A
D
E
,
[
V
I
.
1
]
therefore also, as the triangle
B
D
E
is to the triangle
A
D
E
, so is the triangle
C
D
E
to the triangle
A
D
E
.
[
V
.
1
1
]
Therefore each of the triangles
B
D
E
,
C
D
E
has the same ratio to
A
D
E
.
Therefore the triangle
B
D
E
is equal to the triangle
C
D
E
;
[
V
.
9
]
and they are on the same base
D
E
.
But equal triangles which are on the same base are also in the same parallels.
[
I
.
3
9
]
Therefore
D
E
is parallel to
B
C
.
Classes
Euclid's Elements
MathWorld
Theorems
Triangles
EuclidBook6
MathWorld
Triangle
