British Flag Theorem
The British flag theorem says that if a point P is chosen inside rectangle ABCD then 
.
![[asy] usepackage("amsmath"); size(200); pair A,B,C,D,P; A=(0,0); B=(200,0); C=(200,150); D=(0,150); P=(124,85); draw(A--B--C--D--cycle); label("<math>A</math>",A,(-1,0)); dot(A); label("<math>B</math>",B,(0,-1)); dot(B); label("<math>C</math>",C,(1,0)); dot(C); label("<math>D</math>",D,(0,1)); dot(D); dot(P); label("<math>P</math>",P,(1,1)); draw((0,85)--(200,85)); draw((124,0)--(124,150)); label("<math>w</math>",(124,0),(0,-1)); label("<math>x</math>",(200,85),(1,0)); label("<math>y</math>",(124,150),(0,1)); label("<math>z</math>",(0,85),(-1,0)); dot((124,0)); dot((200,85)); dot((124,150)); dot((0,85)); [/asy]](http://latex.artofproblemsolving.com/c/9/0/c9003629a7690ea13dfae7e27c580f8bf1293bab.png)
The theorem also applies to points outside 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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