Easy identity

by shiningsunnyday, Oct 18, 2016, 11:25 AM

Well-known / 105 Algebra Problems wrote:

Prove that $(x_1^2+x_2^2+ \ldots + x_n^2)(y_1^2+y_2^2+ \ldots + y_n^2) = (x_1y_1 + x_2y_2 + \ldots + x_ny_n)^2 + \sum_{1 \le i < j \le n} (x_iy_j - x_jy_i)^2.$

Solution
Remark
Tidbit

This post has been edited 1 time. Last edited by shiningsunnyday, Oct 18, 2016, 11:27 AM

algebra

Manipulation

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wow nice proof of cauchy

Thanks

This post has been edited 1 time. Last edited by shiningsunnyday, Oct 21, 2016, 8:27 AM

by MathStudent2002, Oct 18, 2016, 11:40 PM

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I feel like the above is tautological (or at least really easy) O wait that's why it's called easy lol.

by FlyingCucumber, Oct 19, 2016, 3:08 AM

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Because the expansion just happens to work that way.

by FlyingCucumber, Oct 19, 2016, 3:09 AM

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(FC's humble brag)

D:

This post has been edited 1 time. Last edited by shiningsunnyday, Oct 21, 2016, 8:28 AM

by zephyrcrush78, Oct 19, 2016, 4:01 AM

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2021 post
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Let $ABC$ be an equilateral triangle of side length $1$ . Let $D$ be the point such that $C$ is the midpoint of $BD$ , and let $I$ be the incenter of triangle $ACD$ . Let $E$ be the point on line $AB$ such that $DE$ and $BI$ are perpendicular. $ \
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all contrib requests added I usually get to posting every night after running on the treadmill and showering, posting my favorite problem(s) I did during the day.
by shiningsunnyday, Jun 16, 2016, 4:37 AM
Since problems is plural, he might post multiple problems through different posts in one day...
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Problems of the day

Easy identity

by shiningsunnyday, Oct 18, 2016, 11:25 AM

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