Set of the British flag

The theorem of the British flag states that the red area is the same as the blue area

Set of the British flag in space:
red area = blue area

The theorem of the British flag (English. British flag theorem ) is a statement in geometry about the connecting lines of a point to the four corners of a rectangle. It says that the two sums of the squared connecting lines to diagonally opposite corner points are equal.

statement

For a rectangle with an interior point , the following equation applies: ${\displaystyle ABCD}$ ${\displaystyle P}$

${\displaystyle |AP|^{2}+|PC|^{2}=|BP|^{2}+|PD|^{2}}$

The sentence remains valid if the point lies outside the rectangle or on its edge. If the point lies on one of the corner points of the rectangle, the equation says that the area of ​​the diagonal square of the rectangle is equal to the sum of the areas of the two side squares of the rectangle, this is the Pythagorean theorem . The theorem of the British flag can thus also be understood as a generalization of the Pythagorean theorem. In addition, the statement is also retained when the point is moved out of the plane into space, that is, the above equation applies to a rectangle and any point in space. ${\displaystyle P}$ ${\displaystyle P}$ ${\displaystyle P}$

proof

Sketch for proof

The proof results directly from multiple applications of the Pythagorean theorem. First you drop perpendiculars from the point on all four sides of the rectangle (see drawing). The links , , and are now hypotenuse of right triangles, in which one can apply the Pythagorean theorem, so we get: ${\displaystyle P}$ ${\displaystyle AP}$ ${\displaystyle PC}$ ${\displaystyle BP}$ ${\displaystyle PD}$

${\displaystyle {\begin{aligned}|AP|^{2}+|PC|^{2}&=(a^{2}+c^{2})+(b^{2}+d^{2})\\&=(a^{2}+d^{2})+(b^{2}+c^{2})\\&=|BP|^{2}+|PD|^{2}\end{aligned}}}$

For the cases where the point is not inside the rectangle, similar proofs result from multiple applications of the Pythagorean theorem.

Naming

The Union Jack , the flag of the United Kingdom .

The name of the sentence goes back to the fact that the drawing in the sketch is reminiscent of the British flag as evidence.

literature

• Nguyen Minh Ha, Dao Thanh Oai: An interesting application of the British flag theorem . Global Journal of Advanced Research on Classical and Modern Geometries, Volume 4 (2015), Issue 1, pp. 31-34.
• Martin Gardner , Dana Richards (Eds.): The Colossal Book of Short Puzzles and Problems . WW Norton, 2006, ISBN 978-0-393-06114-7 , pp. 147, 159 (problem 6.16)
• Euclid , Dionysius Lardner (Ed.): The First Six Books of the Elements of Euclid . John Taylor, London 1828, p. 87 ( online copy in the Internet Archive )

Web links

Commons : British flag theorem  - collection of images, videos and audio files

• British Flag Theorem on artofproblemsolving.com
• Can You Solve Microsoft's Rectangle Corners Interview Question? (Video, English, 5:41 min.)
• Puzzle # 170: Rectangle and the Oil Well - application of the theorem from the British flag, corresponds to problem 6.16 at Gardner
• Harvard-MIT Mathematics Tournament 2003 - Problem 28 is an application of the theorem of the British flag