MathDB

3

Part of 1999 Dutch Mathematical Olympiad

Problems(1)

Rotating a square + Projection on a line

Source: Dutch NMO 1999

10/22/2005
Let ABCDABCD be a square and let ℓ\ell be a line. Let MM be the centre of the square. The diagonals of the square have length 2 and the distance from MM to ℓ\ell exceeds 1. Let A′,B′,C′,D′A',B',C',D' be the orthogonal projections of A,B,C,DA,B,C,D onto ℓ\ell. Suppose that one rotates the square, such that MM is invariant. The positions of A,B,C,D,A′,B′,C′,D′A,B,C,D,A',B',C',D' change. Prove that the value of AA′2+BB′2+CC′2+DD′2AA'^2 + BB'^2 + CC'^2 + DD'^2 does not change.
rotationcomplex numbers