Difference between revisions of "Geometric inequality"
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===Triangle Inequality=== | ===Triangle Inequality=== | ||
− | The [[Triangle Inequality]] says that the sum of the lengths of any two sides of a non[[degenerate]] triangle is greater than the length of the third side. This inequality is particularly useful and shows up frequently on Intermediate level geometry problems. It also provides the basis for the definition of a [[metric space]] in [[analysis]]. | + | The [[Triangle Inequality]] says that the sum of the lengths of any two sides of a non[[degenerate]] triangle is greater than the length of the third side. This inequality is particularly useful and shows up frequently on Intermediate level geometry problems. It also provides the basis for the definition of a [[metric space]] in [[analysis]]. |
===Pythagorean Inequality=== | ===Pythagorean Inequality=== |
Revision as of 09:17, 23 January 2015
A geometric inequality is an inequality involving various measures (angles, lengths, areas, etc.) in geometry.
Contents
[hide]Triangle Inequality
The Triangle Inequality says that the sum of the lengths of any two sides of a nondegenerate triangle is greater than the length of the third side. This inequality is particularly useful and shows up frequently on Intermediate level geometry problems. It also provides the basis for the definition of a metric space in analysis.
Pythagorean Inequality
The Pythagorean Inequality is a generalization of the Pythagorean Theorem. The Theorem states that in a right triangle with sides of length we have . The Inequality extends this to obtuse and acute triangles. The inequality says:
For an acute triangle with sides of length , . For an obtuse triangle with sides , .
This inequality is a direct result of the Law of Cosines, although it is also possible to prove without using trigonometry.
Isoperimetric Inequality
The Isoperimetric Inequality states that 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.
Trigonometric Inequalities
- In , .
Proof: is a concave function from . Therefore we may use Jensen's inequality:
Alternatively, we may use a method that can be called "perturbation". If we let all the angles be equal, we prove that if we make one angle greater and the other one smaller, we will decrease the total value of the expression. To prove this, all we need to show is if , then . This inequality reduces to , which is equivalent to . Since this is always true for , this inequality is true. Therefore, the maximum value of this expression is when , which gives us the value .
Similarly, in , .
Euler's inequality
Euler's inequality states that , where and denote the circumradius and inradius of triangle , respectively.
Erdos-Mordell inequality
The Erdős–Mordell inequality states that for any triangle ABC and point P inside ABC, the sum of the distances from P to the sides is less than or equal to half of the sum of the distances from P to the vertices. This article is a stub. Help us out by expanding it.