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Euclid Book 6
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Euclid Book 6 Proposition 15b
Statement
T
w
o
t
r
i
a
n
g
l
e
s
(
△
A
C
B
,
△
D
F
E
)
w
i
t
h
a
p
a
i
r
o
f
e
q
u
a
l
a
n
g
l
e
s
(
∠
A
C
B
,
∠
D
F
E
)
,
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
(
C
A
F
D
F
E
C
B
)
,
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
.
}
,
{
}
}
,
T
r
i
a
n
g
l
e
[
{
A
.
,
C
.
,
B
.
}
]
,
T
r
i
a
n
g
l
e
[
{
D
.
,
F
.
,
E
.
}
]
,
P
l
a
n
a
r
A
n
g
l
e
[
{
A
.
,
C
.
,
B
.
}
]
P
l
a
n
a
r
A
n
g
l
e
[
{
D
.
,
F
.
,
E
.
}
]
,
E
u
c
l
i
d
e
a
n
D
i
s
t
a
n
c
e
[
C
.
,
A
.
]
E
u
c
l
i
d
e
a
n
D
i
s
t
a
n
c
e
[
F
.
,
D
.
]
E
u
c
l
i
d
e
a
n
D
i
s
t
a
n
c
e
[
F
.
,
E
.
]
E
u
c
l
i
d
e
a
n
D
i
s
t
a
n
c
e
[
C
.
,
B
.
]
,
{
A
r
e
a
[
T
r
i
a
n
g
l
e
[
{
A
.
,
C
.
,
B
.
}
]
]
A
r
e
a
[
T
r
i
a
n
g
l
e
[
{
D
.
,
F
.
,
E
.
}
]
]
}
A
r
e
a
[
T
r
i
a
n
g
l
e
[
{
A
.
,
C
.
,
B
.
}
]
]
A
r
e
a
[
T
r
i
a
n
g
l
e
[
{
D
.
,
F
.
,
E
.
}
]
]
Explanations
Let the sides of the triangles
A
B
C
,
A
D
E
be reciprocally proportional, that is to say, let
E
A
be to
A
B
as
C
A
to
A
D
; I say that the triangle
A
B
C
is equal to the triangle
A
D
E
.
For, if
B
D
be again joined, since, as
C
A
is to
A
D
, so is
E
A
to
A
B
, while, as
C
A
is to
A
D
, so is the triangle
A
B
C
to the triangle
B
A
D
, and, as
E
A
is to
A
B
, so is the triangle
E
A
D
to the triangle
B
A
D
,
[
V
I
.
1
]
therefore, as the triangle
A
B
C
is to the triangle
B
A
D
, so is the triangle
E
A
D
to the triangle
B
A
D
.
[
V
.
1
1
]
Therefore each of the triangles
A
B
C
,
E
A
D
has the same ratio to
B
A
D
.
Therefore the triangle
A
B
C
is equal to the triangle
E
A
D
.
[
V
.
9
]
Classes
Euclid's Elements
Theorems
Triangles
EuclidBook6
Related Theorems
EuclidBook6Proposition14b
EuclidBook6Proposition15a
