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Difference between revisions of "Geometric inequality"

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===Isoperimetric Inequality===
 
===Isoperimetric Inequality===
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The [[Isoperimetric Inequality]] states that if a figure in the plane has [[area]] <math>A</math> and [[perimeter]] <math>P</math>, then <math>\frac{4\pi A}{p^2} < 1</math>. This means that given a perimeter <math>P</math> for a plane figure, the [[circle]] has the largest area. Conversely, of all plane figures with area <math>A</math>, the circle has the least perimeter.
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The [[Isoperimetric Inequality]] states that if a figure in the plane has [[area]] <math>A</math> and [[perimeter]] <math>P</math>, then <math>\frac{4\pi A}{p^2} \le 1</math>. This means that given a perimeter <math>P</math> for a plane figure, the [[circle]] has the largest area. Conversely, of all plane figures with area <math>A</math>, the circle has the least perimeter.
  
 
===Trigonometric Inequalities===
 
===Trigonometric Inequalities===
 
*In <math>\triangle ABC</math>, <math>\sin{A}+\sin{B}+\sin{C}\le \frac{3\sqrt{3}}{2}</math>. Proof: <math>\sin</math> is a [[concave function]] from <math>0\le \theta \le \pi</math>. Therefore we may use [[Jensen's inequality]]: <math>\frac{\sin{A}+\sin{B}+\sin{C}}{3}\le \sin{\left(\frac{A+B+C}{3}\right)}=\frac{\sqrt{3}}{2}</math>
 
*In <math>\triangle ABC</math>, <math>\sin{A}+\sin{B}+\sin{C}\le \frac{3\sqrt{3}}{2}</math>. Proof: <math>\sin</math> is a [[concave function]] from <math>0\le \theta \le \pi</math>. Therefore we may use [[Jensen's inequality]]: <math>\frac{\sin{A}+\sin{B}+\sin{C}}{3}\le \sin{\left(\frac{A+B+C}{3}\right)}=\frac{\sqrt{3}}{2}</math>

Revision as of 23:09, 28 April 2008

A geometric inequality is an inequality involving various measures (angles, lengths, areas, etc.) in geometry.

Pythagorean Inequality

The Pythagorean Inequality is a generalization of the Pythagorean Theorem. The Theorem states that in a right triangle with sides of length $a \leq b \leq c$ we have $a^2 + b^2 = c^2$. The Inequality extends this to obtuse and acute triangles. The inequality says:

For an acute triangle with sides of length $a \leq b \leq c$, $a^2+b^2>c^2$. For an obtuse triangle with sides $a \leq b \leq c$, $a^2+b^2<c^2$.

This inequality is a direct result of the Law of Cosines, although it is also possible to prove without using trigonometry.

Triangle Inequality

The Triangle Inequality says that the sum of the lengths of any two sides of a nondegenerate triangle is greater than the length of the third side. This inequality is particularly useful and shows up frequently on Intermediate level geometry problems. It also provides the basis for the definition of a metric space in analysis.

Isoperimetric Inequality

The Isoperimetric Inequality states that if a figure in the plane has area $A$ and perimeter $P$, then $\frac{4\pi A}{p^2} \le 1$. This means that given a perimeter $P$ for a plane figure, the circle has the largest area. Conversely, of all plane figures with area $A$, the circle has the least perimeter.

Trigonometric Inequalities

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