Under Development
A collection of classical geometry in computable formats along with code and diagrams.
Computable Euclid
›
Euclid Book 5
›
Browse books
Euclid Book 1
Euclid Book 2
Euclid Book 3
Euclid Book 4
Euclid Book 5
Euclid Book 6
Euclid Book 13
Euclid Book 5 Proposition 21
Statement
G
i
v
e
n
t
w
o
g
r
o
u
p
s
o
f
t
h
r
e
e
m
a
g
n
i
t
u
d
e
s
(
A
B
,
C
D
,
a
n
d
E
F
;
G
H
,
I
J
,
a
n
d
K
L
)
,
i
f
t
h
e
f
i
r
s
t
m
a
g
n
i
t
u
d
e
i
s
t
o
t
h
e
s
e
c
o
n
d
a
s
t
h
e
f
i
f
t
h
i
s
t
o
t
h
e
s
i
x
t
h
(
A
B
C
D
I
J
K
L
)
,
t
h
e
s
e
c
o
n
d
i
s
t
o
t
h
e
t
h
i
r
d
a
s
t
h
e
f
o
u
r
t
h
i
s
t
o
t
h
e
f
i
f
t
h
(
C
D
E
F
G
H
I
J
)
,
a
n
d
t
h
e
f
i
r
s
t
m
a
g
n
i
t
u
d
e
i
s
g
r
e
a
t
e
r
t
h
a
n
o
r
e
q
u
a
l
t
o
t
h
e
t
h
i
r
d
(
A
B
≥
E
F
)
,
t
h
e
n
t
h
e
f
o
u
r
t
h
m
a
g
n
i
t
u
d
e
i
s
a
l
s
o
g
r
e
a
t
e
r
t
h
a
n
o
r
e
q
u
a
l
t
o
t
h
e
s
i
x
t
h
(
G
H
≥
K
L
)
.
Computational Explanation
Explanations
Let there be three magnitudes
A
,
B
,
C
, and others
D
,
E
,
F
equal to them in multitude, which taken two and two are in the same ratio, and let the proportion of them be perturbed, so that, as
A
is to
B
, so is
E
to
F
, and, as
B
is to
C
, so is
D
to
E
, and let
A
be greater than
C
ex aequali; I say that
D
will also be greater than
F
; if
A
is equal to
C
, equal; and if less, less.
For, since
A
is greater than
C
, and
B
is some other magnitude, therefore
A
has to
B
a greater ratio than
C
has to
B
.
[
V
.
8
]
But, as
A
is to
B
, so is
E
to
F
, and, as
C
is to
B
, inversely, so is
E
to
D
. Therefore also
E
has to
F
a greater ratio than
E
has to
D
.
[
V
.
1
3
]
But that to which the same has a greater ratio is less;
[
V
.
1
0
]
therefore
F
is less than
D
; therefore
D
is greater than
F
.
Similarly we can prove that, if
A
be equal to
C
,
D
will also be equal to
F
; and if less, less.
Classes
Euclid's Elements
Theorems
Geometric Algebra
EuclidBook5
Related Theorems
EuclidBook5Proposition20
EuclidBook5Proposition22
EuclidBook5Proposition23
