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British Flag Theorem

From AoPSWiki

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

[Asy_image]

The theorem also applies to points outside the rectangle, 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:

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

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