Geometric inequality

Revision as of 17:21, 20 June 2006 by Cosinator (talk | contribs) (Pythagorean Inequality: Correcting spelling)

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.

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