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Euclid Book 13
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Euclid Book 13 Proposition 12
Statement
Computational Explanation
Explanations
Let
A
B
C
be a circle, and let the equilateral triangle
A
B
C
be inscribed in it; I say that the square on one side of the triangle
A
B
C
is triple of the square on the radius of the circle.
For let the centre
D
of the circle
A
B
C
be taken, let
A
D
be joined and carried through to
E
, and let
B
E
be joined.
Then, since the triangle
A
B
C
is equilateral, therefore the circumference
B
E
C
is a third part of the circumference of the circle
A
B
C
.
Therefore the circumference
B
E
is a sixth part of the circumference of the circle; therefore the straight line
B
E
belongs to a hexagon; therefore it is equal to the radius
D
E
.
[
I
V
.
1
5
]
And, since
A
E
is double of
D
E
, the square on
A
E
is quadruple of the square on
E
D
, that is, of the square on
B
E
.
But the square on
A
E
is equal to the squares on
A
B
,
B
E
;
[
I
I
I
.
3
1
]
[
I
.
4
7
]
therefore the squares on
A
B
,
B
E
are quadruple of the square on
B
E
.
Therefore, separando, the square on
A
B
is triple of the square on
B
E
.
But
B
E
is equal to
D
E
; therefore the square on
A
B
is triple of the square on
D
E
.
Therefore the square on the side of the triangle is triple of the square on the radius.
Classes
Euclid's Elements
Theorems
EuclidBook13
