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 a^2+b^2=c^2 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, 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 [[law of cosines|The Law of Cosines]] which always attains equality. | + | 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=== | ||
Revision as of 12:54, 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 
for right triangles. The Inequality extends this to obtuse and acute triangles. The inequality says:
For acute triangles, 
. For obtuse triangles, 
. 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 
and perimeter 
then 
. This means that given a perimeter for a plane figure, the circle has the largest area. Conversely, of all plane figures with area the circle has the least perimeter.
