|
|
| Line 1: |
Line 1: |
| [[File:British flag theorem squares.svg|thumb|240px|According to the British flag theorem, the red squares have the same total area as the blue squares]]
| | Golda is what's created on my beginning certificate even though it is not the title on my beginning certificate. Doing ballet is something she would by no means give up. He is an information officer. Some time in the past she chose to reside in Alaska and her mothers and fathers reside nearby.<br><br>Also visit my blog post; online psychic chat ([http://Bigpolis.com/blogs/post/6503 Learn Additional]) |
| In [[Euclidean geometry]], the '''British flag theorem''' says that if a point ''P'' is chosen inside [[rectangle]] ''ABCD'' then the sum of the squared [[Euclidean distance]]s from ''P'' to two opposite corners of the rectangle equals the sum to the other two opposite corners.<ref>{{citation|title=The First Six Books of the Elements of Euclid|first=Dionysius|last=Lardner|authorlink=Dionysius Lardner|publisher=H.G. Bohn|year=1848|page=87|url=http://books.google.com/books?id=5INRAAAAYAAJ&pg=PA87}}. Lardner includes this theorem in what he calls "the most useful and remarkable theorems which may be inferred" from the results in Book II of [[Euclid's Elements]].</ref><ref>{{citation|title=Elementary Mathematical Analysis|first1=John Wesley|last1=Young|author1-link = John Wesley Young|first2=Frank Millett|last2=Morgan|publisher=The Macmillan company|year=1917|page=304|url=http://books.google.com/books?id=guI3AAAAMAAJ&pg=PA304}}.</ref><ref>{{citation|title=Plane Analytic Geometry: with introductory chapters on the differential calculus|first=Maxime|last=Bôcher|authorlink=Maxime Bôcher |publisher=H. Holt and Company|year=1915|page=17|url=http://books.google.com/books?id=bYkLAAAAYAAJ&pg=PA17}}.</ref>
| |
| As an [[equation]]:
| |
| : <math>AP^{2}+CP^{2}=BP^{2}+DP^{2}.\,</math>
| |
| | |
| 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.<ref>[http://web.mit.edu/hmmt/www/datafiles/solutions/2003/sguts03.pdf Harvard-MIT Mathematics Tournament solutions], Problem 28.</ref> Even more generally, if the sums of squared 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 of the two sums will depend only on the shape of the parallelogram and not on the choice of ''P''.<ref>{{citation|title=Lessons in Geometry: Plane geometry|first=Jacques|last=Hadamard|authorlink=Jacques Hadamard|publisher=American Mathematical Society|year=2008|isbn=978-0-8218-4367-3|page=136|url=http://books.google.com/books?id=fLwydFiM7zMC&pg=PA136}}.</ref>
| |
| | |
| == Proof ==
| |
| [[File:Britishflag.png|thumb|Illustration for proof]]
| |
| Drop [[perpendicular line]]s from the point ''P'' to the sides of the rectangle, meeting sides ''AB'', ''BC'', ''CD'', and ''AD'' in points ''w'', ''x'', ''y'' and ''z'' respectively, as shown in the figure; these four points ''wxyz'' form the vertices of an [[orthodiagonal quadrilateral]].
| |
| By applying the [[Pythagorean theorem]] to the [[right triangle]] ''AwP'', and observing that ''wP'' = ''Az'', it follows that
| |
| * <math>AP^{2} = Aw^{2} + wP^{2} = Aw^{2} + Az^{2}</math>
| |
| and by a similar argument the squared lengths of the distances from ''P'' to the other three corners can be calculated as
| |
| * <math>PC^{2} = wB^{2} + zD^{2},</math>
| |
| * <math>BP^{2} = wB^{2} + Az^{2},</math> and
| |
| * <math>PD^{2} = zD^{2} + Aw^{2}.</math>
| |
| | |
| Therefore:
| |
| | |
| : <math>AP^{2} + PC^{2} = (Aw^{2} + Az^{2}) + (wB^{2} + zD^{2}) = (wB^{2} + Az^{2}) + (zD^{2} + Aw^{2}) = BP^{2} + PD^{2}.\,</math>
| |
| | |
| ==Naming==
| |
| [[File:Flag of the United Kingdom.svg|thumb|The flag of the [[United Kingdom]].]]
| |
| This theorem takes its name from the fact that, when the [[line segment]]s from ''P'' to the corners of the rectangle are drawn, together with the perpendicular lines used in the proof, the completed figure somewhat resembles a [[Union Flag]].
| |
| | |
| == References ==
| |
| {{reflist}}
| |
| | |
| {{DEFAULTSORT:British Flag Theorem}}
| |
| [[Category:Euclidean geometry]]
| |
| [[Category:Theorems in plane geometry]]
| |
Golda is what's created on my beginning certificate even though it is not the title on my beginning certificate. Doing ballet is something she would by no means give up. He is an information officer. Some time in the past she chose to reside in Alaska and her mothers and fathers reside nearby.
Also visit my blog post; online psychic chat (Learn Additional)