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Euclid Book 6
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Euclid Book 6 Proposition 3b
Statement
I
f
t
w
o
s
e
g
m
e
n
t
s
(
B
D
,
D
C
)
o
f
t
h
e
b
a
s
e
(
B
C
)
h
a
v
e
t
h
e
s
a
m
e
r
a
t
i
o
a
s
t
h
e
r
e
m
a
i
n
i
n
g
s
i
d
e
s
(
A
B
,
A
C
)
o
f
t
h
e
t
r
i
a
n
g
l
e
(
△
A
B
C
)
,
t
h
e
n
t
h
e
l
i
n
e
s
e
g
m
e
n
t
(
A
D
)
j
o
i
n
i
n
g
t
h
e
v
e
r
t
e
x
t
o
t
h
e
p
o
i
n
t
o
f
i
n
t
e
r
s
e
c
t
i
o
n
b
i
s
e
c
t
s
t
h
a
t
a
n
g
l
e
(
∠
B
A
C
)
o
f
t
h
e
t
r
i
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
.
}
,
{
A
.
,
D
.
}
}
]
,
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
.
]
,
{
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
.
}
]
}
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
.
}
]
Explanations
Let
B
A
be to
A
C
as
B
D
to
D
C
, and let
A
D
be joined; I say that the angle
B
A
C
has been bisected by the straight line A.
D. For, with the same construction, since, as
B
D
is to
D
C
, so is
B
A
to
A
C
, and also, as
B
D
is to
D
C
, so is
B
A
to
A
E
: for
A
D
has been drawn parallel to
E
C
, one of the sides of the triangle
B
C
E
:
[
V
I
.
2
]
therefore also, as
B
A
is to
A
C
, so is
B
A
to
A
E
.
[
V
.
1
1
]
Therefore
A
C
is equal to
A
E
,
[
V
.
9
]
so that the angle
A
E
C
is also equal to the angle
A
C
E
.
[
I
.
5
]
But the angle
A
E
C
is equal to the exterior angle
B
A
D
,
[
I
.
2
9
]
and the angle
A
C
E
is equal to the alternate angle
C
A
D
;[id.]therefore the angle
B
A
D
is also equal to the angle
C
A
D
.
Therefore the angle
B
A
C
has been bisected by the straight line
A
D
.
Classes
Euclid's Elements
MathWorld
Theorems
Triangles
EuclidBook6
MathWorld
AngleBisectorTheorem
