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Consider two points in the Euclidean plane:

$A=(A_1,A_2),B=(B_1,B_2)\in\mathbb{R}^2$,

and some fixed real number $\lambda\in(0,1)$. The claim is that the following expression is always a positive real number:

$$(1-\lambda^2)\cdot (\vert B \vert^2- \vert A\vert^2)+(A_1+B_1\cdot \lambda^2)^2+(A_2+B_2\cdot \lambda^2)^2>0.$$

I'm wondering why this complicated looking real number is always positive? I plugged in some numbers and it was positive. Is it possible to get the minimum of this function? Best regards.

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  • $\begingroup$ The statement looks in complete. "The claim is that there exists some circle with radius given by" The way you ask your question, makes it sound as though it is just a fancy way of saying it is non negative. $\endgroup$ – grdgfgr May 30 '15 at 5:17
  • $\begingroup$ By the way, there's no square root sign anymore. You might want to edit the question once again to reflect that. $\endgroup$ – wltrup May 30 '15 at 5:29
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Expand that expression and you'll see that it equals $$\lambda^2\,|A+B|^2 + (1-\lambda^2)^2\,|B|^2\,,$$ which is clearly a non-negative quantity.

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