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Euclid Book 6
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Euclid Book 6 Proposition 31
Statement
Computational Explanation
Explanations
Let
A
B
C
be a right-angled triangle having the angle
B
A
C
right; I say that the figure on
B
C
is equal to the similar and similarly described figures on
B
A
,
A
C
.
Let
A
D
be drawn perpendicular.
Then since, in the right-angled triangle
A
B
C
,
A
D
has been drawn from the right angle at
A
perpendicular to the base
B
C
, the triangles
A
B
D
,
A
D
C
adjoining the perpendicular are similar both to the whole
A
B
C
and to one another.
[
V
I
.
8
]
And, since
A
B
C
is similar to
A
B
D
, therefore, as
C
B
is to
B
A
, so is
A
B
to
B
D
.
[
V
I
.
D
e
f
.
1
]
And, since three straight lines are proportional, as the first is to the third, so is the figure on the first to the similar and similarly described figure on the second.
[
V
I
.
1
9
]
Therefore, as
C
B
is to
B
D
, so is the figure on
C
B
to the similar and similarly described figure on
B
A
.
For the same reason also, as
B
C
is to
C
D
, so is the figure on
B
C
to that on
C
A
; so that, in addition, as
B
C
is to
B
D
,
D
C
, so is the figure on
B
C
to the similar and similarly described figures on
B
A
,
A
C
.
But
B
C
is equal to
B
D
,
D
C
; therefore the figure on
B
C
is also equal to the similar and similarly described figures on
B
A
,
A
C
.
Classes
Euclid's Elements
Theorems
EuclidBook6
