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Euclid Book 6
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Euclid Book 6 Proposition 3a
Statement
I
f
a
l
i
n
e
(
A
D
)
b
i
s
e
c
t
s
a
n
y
a
n
g
l
e
(
∠
B
A
C
)
o
f
a
t
r
i
a
n
g
l
e
(
△
A
B
C
)
,
i
t
d
i
v
i
d
e
s
t
h
e
o
p
p
o
s
i
t
e
s
i
d
e
(
B
C
)
i
n
t
h
e
s
a
m
e
r
a
t
i
o
a
s
t
h
e
s
i
d
e
s
a
d
j
a
c
e
n
t
t
o
t
h
e
a
n
g
l
e
.
Computational Explanation
G
e
o
m
e
t
r
i
c
S
c
e
n
e
{
{
A
.
,
B
.
,
C
.
,
D
.
}
,
{
}
}
,
{
T
r
i
a
n
g
l
e
[
{
A
.
,
B
.
,
C
.
}
]
,
L
i
n
e
[
{
B
.
,
D
.
,
C
.
}
]
,
I
n
f
i
n
i
t
e
L
i
n
e
[
{
A
.
,
D
.
}
]
A
n
g
l
e
B
i
s
e
c
t
o
r
[
{
B
.
,
A
.
,
C
.
}
]
}
,
E
u
c
l
i
d
e
a
n
D
i
s
t
a
n
c
e
[
B
.
,
D
.
]
E
u
c
l
i
d
e
a
n
D
i
s
t
a
n
c
e
[
D
.
,
C
.
]
E
u
c
l
i
d
e
a
n
D
i
s
t
a
n
c
e
[
B
.
,
A
.
]
E
u
c
l
i
d
e
a
n
D
i
s
t
a
n
c
e
[
A
.
,
C
.
]
E
u
c
l
i
d
e
a
n
D
i
s
t
a
n
c
e
[
B
.
,
D
.
]
E
u
c
l
i
d
e
a
n
D
i
s
t
a
n
c
e
[
D
.
,
C
.
]
E
u
c
l
i
d
e
a
n
D
i
s
t
a
n
c
e
[
B
.
,
A
.
]
E
u
c
l
i
d
e
a
n
D
i
s
t
a
n
c
e
[
A
.
,
C
.
]
Explanations
Let
A
B
C
be a triangle, and let the angle
B
A
C
be bisected by the straight line
A
D
; I say that, as
B
D
is to
C
D
, so is
B
A
to
A
C
.
For let
C
E
be drawn through
C
parallel to
D
A
, and let
B
A
be carried through and meet it at
. Then, since the straight line
A
C
falls upon the parallels
A
D
,
E
C
, the angle
A
C
E
is equal to the angle
C
A
D
.
[
I
.
2
9
]
But the angle
C
A
D
is by hypothesis equal to the angle
B
A
D
; therefore the angle
B
A
D
is also equal to the angle
A
C
E
.
Again, since the straight line
B
A
E
falls upon the parallels
A
D
,
E
C
, the exterior angle
B
A
D
is equal to the interior angle
A
E
C
.
[
I
.
2
9
]
But the angle
A
C
E
was also proved equal to the angle
B
A
D
; therefore the angle
A
C
E
is also equal to the angle
A
E
C
, so that the side
A
E
is also equal to the side
A
C
.
[
I
.
6
]
And, since
A
D
has been drawn parallel to
E
C
, one of the sides of the triangle
B
C
E
, therefore, proportionally, as
B
D
is to
D
C
, so is
B
A
to
A
E
.
But
A
E
is equal to
A
C
;
[
V
I
.
2
]
therefore, as
B
D
is to
D
C
, so is
B
A
to
A
C
.
Classes
Euclid's Elements
MathWorld
Theorems
Triangles
EuclidBook6
MathWorld
AngleBisectorTheorem
