Inequalities in quadrilaterals are inequalities that relate different quantities in a quadrilateral . The inequalities apply if the square is in (uncurved) Euclidean space . in the following denote the side lengths, the diagonal lengths of a square.
R.
3
{\ displaystyle \ mathbb {R} ^ {3}}
a
,
b
,
c
,
d
{\ displaystyle a, b, c, d}
e
,
f
{\ displaystyle e, f}
Generalized triangle inequality
In each square, the sum of any three side lengths is greater than the fourth side length:
a
+
b
+
c
>
d
,
b
+
c
+
d
>
a
,
a
+
c
+
d
>
b
,
a
+
b
+
d
>
c
.
{\ displaystyle a + b + c> d, \ quad b + c + d> a, \ quad a + c + d> b, \ quad a + b + d> c.}
It follows:
a
2
+
b
2
+
c
2
>
d
2
3
{\ displaystyle a ^ {2} + b ^ {2} + c ^ {2}> {\ frac {d ^ {2}} {3}}}
Ptolemaic inequality
In every square it holds
a
⋅
c
+
b
⋅
d
≥
e
⋅
f
{\ displaystyle a \ cdot c + b \ cdot d \ geq e \ cdot f}
.
In the case of a chordal quadrilateral , equality applies ( Ptolemy's theorem ).
Inequality between circumference and diagonals
In every convex quadrilateral the sum of the diagonal lengths lies between half and the whole circumference :
1
2
(
a
+
b
+
c
+
d
)
<
e
+
f
<
a
+
b
+
c
+
d
{\ displaystyle {\ frac {1} {2}} (a + b + c + d) <e + f <a + b + c + d}
Square inequality for metrics
The square inequality in metric space follows from the triangle inequality :
|
d
(
x
,
y
)
-
d
(
u
,
v
)
|
≤
d
(
x
,
u
)
+
d
(
v
,
y
)
{\ displaystyle {\ Big |} d (x, y) -d (u, v) {\ Big |} \ leq d (x, u) + d (v, y)}
.
Proof:
Applying the triangle inequality multiple times we get:
d
(
x
,
y
)
≤
d
(
x
,
u
)
+
d
(
u
,
v
)
+
d
(
v
,
y
)
{\ displaystyle d (x, y) \ leq d (x, u) + d (u, v) + d (v, y)}
or.
d
(
u
,
v
)
≤
d
(
u
,
x
)
+
d
(
x
,
y
)
+
d
(
y
,
v
)
{\ displaystyle d (u, v) \ leq d (u, x) + d (x, y) + d (y, v)}
Using the properties of metrics and absolute amounts then applies
|
d
(
x
,
y
)
-
d
(
u
,
v
)
|
=
d
(
x
,
y
)
-
d
(
u
,
v
)
≤
d
(
x
,
u
)
+
d
(
u
,
v
)
+
d
(
v
,
y
)
-
d
(
u
,
v
)
=
d
(
x
,
u
)
+
d
(
v
,
y
)
{\ displaystyle {\ Big |} d (x, y) -d (u, v) {\ Big |} = d (x, y) -d (u, v) \ leq d (x, u) + d (u, v) + d (v, y) -d (u, v) = d (x, u) + d (v, y)}
if applies or in case
d
(
x
,
y
)
-
d
(
u
,
v
)
≥
0
{\ displaystyle d (x, y) -d (u, v) \ geq 0}
d
(
x
,
y
)
-
d
(
u
,
v
)
≤
0
{\ displaystyle d (x, y) -d (u, v) \ leq 0}
|
d
(
x
,
y
)
-
d
(
u
,
v
)
|
=
d
(
u
,
v
)
-
d
(
x
,
y
)
≤
d
(
u
,
x
)
+
d
(
x
,
y
)
+
d
(
y
,
v
)
-
d
(
x
,
y
)
=
d
(
x
,
u
)
+
d
(
v
,
y
)
{\ displaystyle {\ Big |} d (x, y) -d (u, v) {\ Big |} = d (u, v) -d (x, y) \ leq d (u, x) + d (x, y) + d (y, v) -d (x, y) = d (x, u) + d (v, y)}
See also
Web links
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