British Flag Theorem
The British flag theorem says that if a point P is chosen inside rectangle ABCD then . The theorem is called the British flag theorem due to the similarities between the British flag and a diagram of the points (shown below):
The theorem also applies if the point is selected outside or on the boundary of the rectangle, although the proof is harder to visualize in this case.
Proof
Squares of lengths suggest right triangles. We build right triangles by drawing a line through perpendicular to two sides of the rectangle, as shown below. Both
and
are rectangles.
Applying the Pythagorean Theorem to each of the four right triangles in the diagram, we have
So, we have
From rectangles
and
, we have
and
, so the expressions above for
and
are equal, as desired.
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