I'm having trouble to show the following inequality:

a, b, c, d, e, f are all some real numbers

sqrt((a-c)^2+(b-d)^2) <= sqrt((a-e)^2+(b-f)^2) + sqrt((c-e)^2+(d-f)^2)

Thanks in advance~

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- Dec 1st 2007, 02:52 PMleojeiQuestion on proving distance formula is metric
I'm having trouble to show the following inequality:

a, b, c, d, e, f are all some real numbers

sqrt((a-c)^2+(b-d)^2) <= sqrt((a-e)^2+(b-f)^2) + sqrt((c-e)^2+(d-f)^2)

Thanks in advance~ - Dec 1st 2007, 03:10 PMJhevon
isn't this just the triangle inequality? the left side is the distance between (a,b) and (c,d) while the right is the distance between (a,b) and (e,f) plus the distance between (e,f) and (c,d). must you prove the triangle inequality?

the easiest way i know is to use vectors (and their magnitudes):

Let the vector beginning at (a,b) and terminating at (c,d) be $\displaystyle a$

Let the vector beginning at (c,d) and terminating at (e,f) be $\displaystyle b$

Then, the vector beginning at (a,b) and terminating at (e,f) is $\displaystyle a + b$

thus we need to prove: $\displaystyle |a + b| \le |a| + |b|$

we can do this by proving: $\displaystyle |a + b|^2 \le (|a|+|b|)^2$

can you continue?

(draw a diagram to make sure my vectors match up with your points, if it doesn't, rename the vectors) - Dec 1st 2007, 03:23 PMleojeiEasiest way indeed
This is the easiest way indeed.

FYI, the original question is this:

Show d[a+bi, c+di]=sqrt((a-c)^2+(b-d)^2) is metric.

Really thanks alot for the idea of using vector. I've been doing Chaos and other Dynamical Systems questions and this question really throw off on me. Thanks~