Theorem

British flag theorem

Geometry • Quadrilateral

Formula

O A^{2} + O C^{2} = O B^{2} + O D^{2}

For a rectangle

A B C D

and any point

O

in its plane,

O A^{2} + O C^{2} = O B^{2} + O D^{2}

, the two pairs being the opposite corners of the rectangle; the point need not lie inside the rectangle for this to hold.

How to use it

1Use it to get a fourth distance when the other three are known: $O D^{2} = O A^{2} + O C^{2} - O B^{2}$ .
2To rebuild it, drop perpendiculars from $O$ to two adjacent sides; the four distances are then hypotenuses built from the same four legs, and both sums collect all four squares once.
3Pair the corners diagonally: $A$ with $C$ and $B$ with $D$ , never $A$ with $B$ .

Worked example

O

lies inside rectangle

A B C D

with

O A = 3

O B = 4

and

O C = 5

. Find

O D

Step-by-step

1
$O A^{2} + O C^{2} = O B^{2} + O D^{2}$
Why: $A, C$ and $B, D$ are the two pairs of opposite corners.
2
$9 + 25 = 16 + O D^{2} \Rightarrow O D^{2} = 18$
3
$O D = 32$

Answer

32

Watch out

The quadrilateral must be a rectangle: in a general parallelogram the two sums differ. Note also that it is a sum, not a difference, and that no coefficient appears on either term.

Builds on

Pythagoras' theorem

Also known as

British flag theorem, point inside a rectangle distances, sum of squares of distances to opposite corners, OA^2 + OC^2 = OB^2 + OD^2

Appears in past papers

2017 (II)

Q69