British flag theorem

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. Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria. In Geometry, a rectangle is defined as a Quadrilateral where all four of its angles are Right angles A rectangle with vertices ABCD would be denoted as

Naming

The flag of the United Kingdom, also known as a British jack. The United Kingdom of Great Britain and Northern Ireland, commonly known as the United Kingdom, the UK or Britain,is a Sovereign state located

This theorem is named as such because when the involved line segments are drawn, they somewhat resemble a British jack. In Geometry, a line segment is a part of a line that is bounded by two distinct end points, and contains every point on the line between its end points The Union Flag, also known as the Union Jack, is the national flag of the United Kingdom of Great Britain and Northern Ireland.

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:

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

References

In Mathematics, the Pythagorean theorem ( American English) or Pythagoras' theorem ( British English) is a relation in Euclidean geometry

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