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Concept# British flag theorem

Summary

In Euclidean geometry, the British flag theorem says that if a point P is chosen inside a rectangle ABCD then the sum of the squares of the Euclidean distances from P to two opposite corners of the rectangle equals the sum to the other two opposite corners.
As an equation:
: AP^2 + CP^2 = BP^2 + DP^2.,
The theorem also applies to points outside the rectangle, and more generally to the distances from a point in Euclidean space to the corners of a rectangle embedded into the space. Even more generally, if the sums of squares of distances from a point P to the two pairs of opposite corners of a parallelogram are compared, the two sums will not in general be equal, but the difference between the two sums will depend only on the shape of the parallelogram and not on the choice of P.
The theorem can also be thought of as a generalisation of the Pythagorean theorem. Placing the point P on any of the four vertices of the rectangle yields the square of the diagonal of the rec

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