Difference between revisions of "Geometric inequality"

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===Pythagorean Inequality===
 
===Pythagorean Inequality===
 
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:
 
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:
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 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.
  
 
===Triangle Inequality===
 
===Triangle Inequality===
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===Isoperimetric Inequality===
 
===Isoperimetric Inequality===
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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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 12:58, 21 June 2006

A Geometric inequality is an inequality involving various measures in geometry.

Pythagorean Inequality

The Pythagorean inequality is the generalization of the Pythagorean Theorem. The Theorem states that $a^2 + b^2 = c^2$ for right triangles. The Inequality extends this to obtuse and acute triangles. The inequality says: For acute triangles, $a^2+b^2>c^2$. For obtuse triangles, $a^2+b^2<c^2$. 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 The Law of Cosines, which always attains equality.

Triangle Inequality

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.


Isoperimetric Inequality

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.