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