British flag theorem

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In Euclidean geometry, the British flag theorem says that if a point P is chosen inside rectangle ABCD then

AP^{2}+PC^{2}=BP^{2}+DP^{2}.\,

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

[edit] Naming

The flag of the United Kingdom, also known as a British jack.
The flag of the United Kingdom, also known as a British jack.

This theorem is named as such because when the involved line segments are drawn, they somewhat resemble a British jack.

[edit] Proof

In Figure 1, by Pythagoras 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}\,

[edit] References

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