# British Flag Theorem

This theorem is famous for Math Olympians but never heard inside our classroom. I’m happy to share this to my readers for you to know this very interesting theorem.

Given a rectangle *ABCD*, we choose point P inside the rectangle and the distance from point A to P is a, B to P is b, C to P is c and D to P is d.

Then,

**Sample Problem:**

1. Let *ABCD* be a rectangle and let *P* be a point inside the rectangle. If *PA **= 8, PB=4* and *PD **= 7*, then what is the length of PC?

Solution:

*PA=a, PC=c, PB=b, PD=d*

Using the formula,

Therefore, PC=1.

2. ABCD is a rectangle and P is inside the rectangle. AP=x-1, BP=x+3, PC=x+2, PD=x-4. How long is BP?

Solution:

*PA=a, PC=c, PB=b, PD=d*

* *Using the formula,

*(x-1) ^{2}+(* x+2)

^{2}=( x+3)

^{2}+( x-4)

^{2}

Simplify and solve for x we have,

x=5

BP=x+2

=5+2

BP**=7**

### Dan

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