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In the case of right triangles, the triangle inequality specializes to the statement that the hypotenuse is greater than either of the two sides and less than their sum. [9] The second part of this theorem is already established above for any side of any triangle. The first part is established using the lower figure.
The basic triangle inequality is +, +, + or equivalently (,,).. In addition, + + + + + <, where the value of the right side is the lowest possible bound, [1]: p. 259 approached asymptotically as certain classes of triangles approach the degenerate case of zero area.
Triangle inequality: If a, b, and c are the lengths of the sides of a triangle then the triangle inequality states that +, with equality only in the degenerate case of a triangle with zero area. In Euclidean geometry and some other geometries, the triangle inequality is a theorem about vectors and vector lengths :
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite the right angle ) is equal to the sum of the areas of the squares on the other two sides.
The reverse inequality follows from the same argument as the standard Minkowski, but uses that Holder's inequality is also reversed in this range. Using the Reverse Minkowski, we may prove that power means with p ≤ 1 , {\textstyle p\leq 1,} such as the harmonic mean and the geometric mean are concave.
Erdős–Mordell inequality. Let be an arbitrary point P inside a given triangle , and let , , and be the perpendiculars from to the sides of the triangles. (If the triangle is obtuse, one of these perpendiculars may cross through a different side of the triangle and end on the line supporting one of the sides.)
In this example, the triangle's side lengths and area are integers, making it a Heronian triangle. However, Heron's formula works equally well when the side lengths are real numbers. As long as they obey the strict triangle inequality, they define a triangle in the Euclidean plane whose area is a positive real number.
In geometry, Barrow's inequality is an inequality relating the distances between an arbitrary point within a triangle, the vertices of the triangle, and certain points on the sides of the triangle. It is named after David Francis Barrow .