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I am working through http://mathworld.wolfram.com/Point-LineDistance2-Dimensional.html and have become stuck in a number of places, presently it is in turning step 10 into step 11 and the identity

$$|ax_0+by_0+c|=|ax+by-ax_0-by_0|$$

I must be missing something from the previous steps. More help may be needed if and when I progress.

Thanks.

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Well, since $ax+by+c=0$ and $|x|=|-x|$... –  Guess who it is. Aug 22 '11 at 10:12

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up vote 2 down vote accepted

WTS: $|ax_0+by_0+c|=|ax+by-ax_0-by_0|$

The thing to note is that $ax + by + c = 0$, as it's the line we're talking about. So we have that $|ax + by - ax_0 - by_0| = |ax + by + c - c - ax_0 - by_0| = |-ax_0 - by_0 - c| = |ax_0 + by_0 + c|$

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Ah, now I see perfectly, stumped by the sub-zeroth stage! Thanks. I found another solution for my needs but nice to learn maths with you. –  John Aug 22 '11 at 12:57

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