# Difference between revisions of "British Flag Theorem"

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− | 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>. | + | 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>. The theorem is called the British flag theorem due to the similarities between the British flag and a diagram of the points (shown below): |

<asy> | <asy> | ||

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P=(124,85); | P=(124,85); | ||

draw(A--B--C--D--cycle); | draw(A--B--C--D--cycle); | ||

+ | draw(A--P); | ||

+ | draw(B--P); | ||

+ | draw(C--P); | ||

+ | draw(D--P); | ||

label("A",A,(-1,0)); | label("A",A,(-1,0)); | ||

dot(A); | dot(A); |

## Revision as of 00:26, 24 January 2010

The **British flag theorem** says that if a point P is chosen inside rectangle ABCD then . The theorem is called the British flag theorem due to the similarities between the British flag and a diagram of the points (shown below):

The theorem also applies if the point is selected outside or on the boundary of the rectangle, although the proof is harder to visualize in this case.

## Proof

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

Therefore:

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