Prove that 1/(a-b)² +1(b-c)²+1/(c-a)² is a perfect square.

1.	Prove that 1/(a-b)² +1(b-c)²+1/(c-a)² is a perfect square.
Answer» <html><body><p><strong>Answer:</strong></p><p>Answer</p><p>Given,</p><p></p><p>a </p><p>2</p><p> +b </p><p>2</p><p> +<a href="https://interviewquestions.tuteehub.com/tag/c-7168" style="font-weight:bold;" target="_blank" title="Click to know more about C">C</a> </p><p>2</p><p> −ab−bc−ca</p><p></p><p>multiply and divide by 2</p><p></p><p>= </p><p>2</p><p>2</p><p> </p><p> ×(a </p><p>2</p><p> +b </p><p>2</p><p> +c </p><p>2</p><p> −ab−bc−ca)</p><p></p><p>= </p><p>2</p><p>a </p><p>2</p><p> −2ab+b </p><p>2</p><p> +b </p><p>2</p><p> −2bc+c </p><p>2</p><p> +c </p><p>2</p><p> −2ac+a </p><p>2</p><p> </p><p> </p><p> </p><p></p><p>= </p><p>2</p><p>(a−b) </p><p>2</p><p> +(b−c) </p><p>2</p><p> +(c−a) </p><p>2</p><p> </p><p> </p><p> </p><p></p><p>square of a number is always <a href="https://interviewquestions.tuteehub.com/tag/greater-476627" style="font-weight:bold;" target="_blank" title="Click to know more about GREATER">GREATER</a> than or equal to zero.</p><p></p><p>∴(a−b) </p><p>2</p><p> +(b−c) </p><p>2</p><p> +(c−a) </p><p>2</p><p> ≥0</p><p></p><p>and </p><p></p><p>(a−b) </p><p>2</p><p> +(b−c) </p><p>2</p><p> +(c−a) </p><p>2</p><p> =0 when a=b=c</p><p></p><p>Hence, a </p><p>2</p><p> +b </p><p>2</p><p> +c </p><p>2</p><p> −ab−bc−ca is always non negative</p></body></html>

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