Geometric inequality: Difference between revisions
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===Triangle Inequality=== | ===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. | 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 <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 14:53, 18 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 trianles, 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.
