Inequalities in quadrilaterals

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. ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle a,b,c,d}$ ${\displaystyle e,f}$

Sizes in a square

Generalized triangle inequality

In each square, the sum of any three side lengths is greater than the fourth side length:

${\displaystyle a+b+c>d,\quad b+c+d>a,\quad a+c+d>b,\quad a+b+d>c.}$

It follows:

${\displaystyle a^{2}+b^{2}+c^{2}>{\frac {d^{2}}{3}}}$

Ptolemaic inequality

In every square it holds

${\displaystyle a\cdot c+b\cdot d\geq e\cdot f}$.

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 :

${\displaystyle {\frac {1}{2}}(a+b+c+d)<e+f<a+b+c+d}$

Square inequality for metrics

The square inequality in metric space follows from the triangle inequality :

${\displaystyle {\Big |}d(x,y)-d(u,v){\Big |}\leq d(x,u)+d(v,y)}$.

Proof:

Applying the triangle inequality multiple times we get:

${\displaystyle d(x,y)\leq d(x,u)+d(u,v)+d(v,y)}$ or.

${\displaystyle d(u,v)\leq d(u,x)+d(x,y)+d(y,v)}$

Using the properties of metrics and absolute amounts then applies

${\displaystyle {\Big |}d(x,y)-d(u,v){\Big |}=d(x,y)-d(u,v)\leq d(x,u)+d(u,v)+d(v,y)-d(u,v)=d(x,u)+d(v,y)}$

if applies or in case ${\displaystyle d(x,y)-d(u,v)\geq 0}$ ${\displaystyle d(x,y)-d(u,v)\leq 0}$

${\displaystyle {\Big |}d(x,y)-d(u,v){\Big |}=d(u,v)-d(x,y)\leq d(u,x)+d(x,y)+d(y,v)-d(x,y)=d(x,u)+d(v,y)}$

See also

• Parallelogram equation

Web links

Wikibooks: Proof of Ptolemy's Theorem  - Learning and Teaching Materials