Matching Pairs British Flag TheoremOnline version Prove: If a point P is chosen inside a rectangle ABCD then the sum of the squares of the Euclidean distances from P to two opposite corners of the rectangle equals the sum of the squares of the Euclidean distances from P to the other two opposite corners of the rectangle. As an equation: ∣AP∣^2+∣CP∣^2=∣BP∣^2+∣DP∣^2 by Özlem Esnaf 1 What if a point P is chosen outside a rectangle ABCD, can we use the theorem to show that the sum of the squares of the Euclidean distances from P to two opposite corners of the rectangle equals to the sum to the other two opposite corners? 2 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 ∣PB∣2=∣WB∣2+∣AZ∣2 and ∣PD∣2=∣ZD∣2+∣AW∣2 3 Therefore; ∣AP∣2+∣CP∣2 = = (∣AW∣2+∣AZ∣2) + (∣WB∣2+∣ZD∣2) =∣AW∣2 + (∣AZ∣2+∣WB∣2) + ∣ZD∣2 =(∣AZ∣2+∣WB∣2) + (∣AW∣2+∣ZD∣2) =∣BP∣2 + ∣DP∣2 4 Drop two lines from the point P to parallel the sides of the rectangle, meeting sides ∣AB∣, ∣BC∣, ∣CD∣, and ∣AD∣ at points W, X, Y and Z respectively. These four points form the vertices of an orthodiagonal quadrilateral WXYZ. 5 How can you show the relationship between the sides of the rectangle WXYZ via this theorem ? By using commutative and associative properties of addition, we can parenthesize the sum of ∣AZ∣2 and ∣WB∣2which equals to ∣BP∣2 and also parenthesize the sum of ∣AW∣2and ∣ZD∣2 which equals to ∣DP∣2. Because the lines ∣WY∣ and ∣XZ∣ are orthogonal to each other, they divide the rectangle ABCD into four rectangles which are AWPZ, ZPYD, PWCY, and WBXP. One of the diagonals of each of them are the lines ∣PA∣, ∣PB∣, ∣PC∣, and ∣PD∣, respectively. In every rectangle, there are two right triangles. Since their hypotenuses are the diagonals of these rectangles, we use these right triangles to find the lengths of the diagonals by applying Pythagorean Theorem. Since the lengths of opposite sides of a rectangle equal to each other, ∣AZ∣ = ∣PW∣ =∣XB∣ ∣ZD∣ = ∣XC∣ = ∣PY∣ ∣AW∣ = ∣ZP∣ = ∣DY∣ ∣WB∣ = ∣PX∣ = ∣YC∣ we can use one of equal sides while writing equations. We can write equations as follows by use of equalities of the sides: ∣AP∣2= ∣AW∣2+∣WP∣2 = ∣AW∣2 +∣AZ∣2 ∣CP∣2 = ∣PX∣2+∣XC∣2 = ∣WB∣2+∣ZD∣2 ∣BP∣2 = ∣PW∣2+∣WB∣2 = ∣AZ∣2+∣WB∣2 ∣DP∣2 = ∣PZ∣2 +∣ZD∣2 = ∣AW∣2+∣ZD∣2 Let’s observe the demonstration in GeoGebra; https://www.geogebra.org/m/K4tBAjc4 (Change the place of the point P.) How can you prove it? (Hint: Use Median Theorem) Since ∣WX∣=∣PB∣, ∣XY∣= ∣PC∣, ∣YZ∣=∣PD∣ and ∣ZW∣ = ∣PA∣, by the theorem the sums of the squares of the length of the opposite sides are equal to each other.
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