Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "British Flag Theorem"
(latex)
Line 2: Line 2:
  
 
<asy>
 
<asy>
−
usepackage("amsmath");
 
 
size(200);
 
size(200);
 
pair A,B,C,D,P;
 
pair A,B,C,D,P;
Line 11: Line 10:
 
P=(124,85);
 
P=(124,85);
 
draw(A--B--C--D--cycle);
 
draw(A--B--C--D--cycle);
−
label("<math>A</math>",A,(-1,0));
+
label("A",A,(-1,0));
 
dot(A);
 
dot(A);
−
label("<math>B</math>",B,(0,-1));
+
label("B",B,(0,-1));
 
dot(B);
 
dot(B);
−
label("<math>C</math>",C,(1,0));
+
label("C",C,(1,0));
 
dot(C);
 
dot(C);
−
label("<math>D</math>",D,(0,1));
+
label("D",D,(0,1));
 
dot(D);
 
dot(D);
 
dot(P);
 
dot(P);
−
label("<math>P</math>",P,(1,1));
+
label("P",P,(1,1));
 
draw((0,85)--(200,85));
 
draw((0,85)--(200,85));
 
draw((124,0)--(124,150));
 
draw((124,0)--(124,150));
−
label("<math>w</math>",(124,0),(0,-1));
+
label("w",(124,0),(0,-1));
−
label("<math>x</math>",(200,85),(1,0));
+
label("x",(200,85),(1,0));
−
label("<math>y</math>",(124,150),(0,1));
+
label("y",(124,150),(0,1));
−
label("<math>z</math>",(0,85),(-1,0));
+
label("z",(0,85),(-1,0));
 
dot((124,0));
 
dot((124,0));
 
dot((200,85));
 
dot((200,85));

Revision as of 16:16, 16 October 2007

The British flag theorem says that if a point P is chosen inside rectangle ABCD then $AP^{2}+PC^{2}=BP^{2}+DP^{2}$.

[asy] 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("A",A,(-1,0)); dot(A); label("B",B,(0,-1)); dot(B); label("C",C,(1,0)); dot(C); label("D",D,(0,1)); dot(D); dot(P); label("P",P,(1,1)); draw((0,85)--(200,85)); draw((124,0)--(124,150)); label("w",(124,0),(0,-1)); label("x",(200,85),(1,0)); label("y",(124,150),(0,1)); label("z",(0,85),(-1,0)); dot((124,0)); dot((200,85)); dot((124,150)); dot((0,85)); [/asy]

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:

  • $AP^{2} = Aw^{2} + Az^{2}$
  • $PC^{2} = wB^{2} + zD^{2}$
  • $BP^{2} = wB^{2} + Az^{2}$
  • $PD^{2} = zD^{2} + Aw^{2}$

Therefore:

  • $AP^{2} + PC^{2} = Aw^{2} + Az^{2} + wB^{2} + zD^{2} = wB^{2} + Az^{2} + zD^{2} + Aw^{2} =\nolinebreak BP^{2} +\nolinebreak PD^{2}$

This article is a stub. Help us out by expanding it.

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=British_Flag_Theorem&oldid=18214"