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Extending the Pythagorean Theorem

Prove that the cube of the hypotenuse of a right triangle is greater than or equal to the sum of the cubes of its other two legs. When does equality hold?

Can you extend this idea to the case of cn?

 

Source: School Science and Mathematics, ?, Problem 4314, p. 106


Hint: Start with the Pythagorean Theorem, and then....

 


Solution Commentary: From Charlie McCracken's solution: If c is the hypotenuse of a right triangle with legs a and b, then c>a and c>b. Multiplying both sides of c2 = a2> + b2 by c gives c3 = ca2> + cb2 > a3> + b3. Equality can only hold in the degenerate case.".....which means what?