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Euclid Book 6
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Euclid Book 6 Proposition 2a
Statement
Computational Explanation
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
