British flag theorem

Drop perpendicular lines from the point P to the sides of the rectangle, meeting sides AB, BC, CD, and AD at points W, X, Y and Z respectively, as shown in the figure. These four points WXYZ form the vertices of an orthodiagonal quadrilateral. By applying the Pythagorean theorem to the right triangle AWP, and observing that WP = AZ, it follows that : AP^2 = AW^2 + WP^2 = AW^2 + AZ^2 and by a similar argument the squares of the lengths of the distances from P to the other three corners can be calculated as : PC^2 = WB^2 + ZD^2, : BP^2 = WB^2 + AZ^2, and : PD^2 = ZD^2 + AW^2. Therefore: :\begin{align} AP^2 + PC^2 &= \left(AW^2 + AZ^2\right) + \left(WB^2 + ZD^2\right) \\[4pt] &= \left(WB^2 + AZ^2\right) + \left(ZD^2 + AW^2\right) \\[4pt] &= BP^2 + PD^2 \end{align} ==Isosceles trapezoid ==

The British flag theorem can be generalized into a statement about (convex) isosceles trapezoids. More precisely for a trapezoid ABCD with parallel sides AB and CD and interior pointP the following equation holds: :|AP|^2+\frac \cdot |PC|^2=|BP|^2+\frac \cdot |PD|^2 In the case of a rectangle the fraction \tfrac evaluates to 1 and hence yields the original theorem. ==Naming==

. This theorem takes its name from the fact that, when the line segments from P to the corners of the rectangle are drawn, together with the perpendicular lines used in the proof, the completed figure resembles a Union Flag. ==See also==