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British Flag Theorem

Revision as of 07:09, 8 October 2007 by 1=2 (talk | contribs) (New page: The '''British flag theorem''' says that if a point P is chosen inside rectangle ABCD then <math>AP^{2}+PC^{2}=BP^{2}+DP^{2}</math>. A---w--------B | | | z---P--------x...)

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The British flag theorem says that if a point P is chosen inside rectangle ABCD then $AP^{2}+PC^{2}=BP^{2}+DP^{2}$ .

 A---w--------B
 |   |        |
 z---P--------x
 |   |        |
 |   |        |
 D---y--------C      
   Figure 1

The theorem also applies to points outside the square, although the proof is harder to visualize in this case.

Proof

In Figure 1, by the Pythagorean theorem, we have:

$AP^{2} = Aw^{2} + Az^{2}$
$PC^{2} = wB^{2} + zD^{2}$
$BP^{2} = wB^{2} + Az^{2}$
$PD^{2} = zD^{2} + Aw^{2}$

Therefore:

$AP^{2} + PC^{2} = Aw^{2} + Az^{2} + wB^{2} + zD^{2} = wB^{2} + Az^{2} + zD^{2} + Aw^{2} = BP^{2} + PD^{2}$

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