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anemone
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Show that for all $a>1$, there is a triangle with sides $a^4-1$, $a^4+a^3+2a^2+a+1$, and $2a^3+a^2+2a+1$.
anemone said:Show that for all $a>1$, there is a triangle with sides $a^4-1$, $a^4+a^3+2a^2+a+1$, and $2a^3+a^2+2a+1$.
DreamWeaver said:Great problem, Anemone! :D
anemone said:Show that for all $a>1$, there is a triangle with sides $a^4-1$, let :$a^4+a^3+2a^2+a+1$, and $2a^3+a^2+2a+1$.
The Triangle Inequality is a mathematical concept that states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
In the given expressions, the Triangle Inequality is applied by comparing the sum of the first two expressions to the third expression. If the sum is greater than the third expression, then the Triangle Inequality is satisfied.
The Triangle Inequality is used to determine whether or not a triangle with sides of length $a^4-1, a^4+a^3+2a^2+a+1, 2a^3+a^2+2a+1$ can exist. If the Triangle Inequality is satisfied, then the lengths of the three sides can form a valid triangle.
Yes, the Triangle Inequality can be applied to any type of triangle, whether it is a right triangle, equilateral triangle, or any other type of triangle.
The Triangle Inequality is a fundamental concept in mathematics and is used in various fields such as geometry, trigonometry, and analysis. It is also used in various proofs and theorems in these areas.