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Difference between revisions of "Geometric inequality"

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A '''geometric inequality''' is an [[inequality]] involving various measures in geometry.
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A '''geometric inequality''' is an [[inequality]] involving various measures ([[angle]]s, [[length]]s, [[area]]s, etc.) in [[geometry]].
  
 
===Pythagorean Inequality===
 
===Pythagorean Inequality===
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The Pythagorean inequality is the generalization of the [[Pythagorean Theorem]]. The Theorem states that <math>a^2 + b^2 = c^2</math> for right triangles. The Inequality extends this to obtuse and acute triangles. The inequality says:
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The Pythagorean Inequality is a generalization of the [[Pythagorean Theorem]]. The Theorem states that in a [[right triangle]] with sides of length <math>a \leq b \leq c</math> we have <math>a^2 + b^2 = c^2</math>. The Inequality extends this to [[obtuse triangle| obtuse]] and [[acute triangle]]s. The inequality says:
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For acute triangles, <math>a^2+b^2>c^2</math>. For obtuse triangles, <math>a^2+b^2<c^2</math>. This fact is easily proven by dropping down altitudes from the triangles, and then doing some algebra to prove that there is an extra segment added.(PROOF added later, once I figure out images). This is a simplified version of [[law of cosines|The Law of Cosines]], which always attains equality.
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For an acute triangle with sides of length <math>a \leq b \leq c</math>, <math>a^2+b^2>c^2</math>. For an obtuse triangle with sides <math>a \leq b \leq c</math>, <math>a^2+b^2<c^2</math>.  
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This inequality is a direct result of the [[Law of Cosines]], although it is also possible to prove without using [[trigonometry]].
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===Triangle Inequality===
 
===Triangle Inequality===
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The Triangle inequality says that the sum of any two sides of a triangle is greater than the third side. This inequality is particularly useful, and shows up frequently on Intermediate level geometry problems.
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The [[Triangle Inequality]] says that the sum of the lengths of any two sides of a non[[degenerate]] 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===
 
===Isoperimetric Inequality===
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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} < 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.

Revision as of 09:38, 14 August 2006

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 $\frac{4\pi A}{p^2} < 1$. 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.

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