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Euclid Book 6
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Euclid Book 6 Proposition 2a
Statement
I
f
a
l
i
n
e
(
D
E
)
i
s
p
a
r
a
l
l
e
l
t
o
a
s
i
d
e
(
B
C
)
o
f
a
t
r
i
a
n
g
l
e
(
△
A
B
C
)
,
a
n
d
i
n
t
e
r
s
e
c
t
i
n
g
w
i
t
h
t
h
e
o
t
h
e
r
s
i
d
e
s
,
t
h
e
n
i
t
d
i
v
i
d
e
s
t
h
o
s
e
s
i
d
e
s
p
r
o
p
o
r
t
i
o
n
a
l
l
y
.
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
.
}
}
]
,
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
"
]
}
,
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
.
]
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
.
]
Explanations
For let
D
E
be drawn parallel to
B
C
, one of the sides of the triangle
A
B
C
; I say that, as
B
D
is to
D
A
, so is
C
E
to
E
A
.
For let
B
E
,
C
D
be joined.
Therefore the triangle
B
D
E
is equal to the triangle
C
D
E
; for they are on the same base
D
E
and in the same parallels
D
E
,
B
C
.
[
I
.
3
8
]
And the triangle
A
D
E
is another area.
But equals have the same ratio to the same;
[
V
.
7
]
therefore, 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
.
But, as the triangle
B
D
E
is to
A
D
E
, so is
B
D
to
D
A
; for, being under the same height, the perpendicular drawn from
to
A
B
, they are to one another as their bases.
[
V
I
.
1
]
For the same reason also, as the triangle
C
D
E
is to
A
D
E
, so is
C
E
to
E
A
. Therefore also, as
B
D
is to
D
A
, so is
C
E
to
E
A
.
[
V
.
1
1
]
Classes
Euclid's Elements
MathWorld
Theorems
Triangles
EuclidBook6
MathWorld
Triangle
