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Quadrilaterals with equal sides

Source: IMO LongList 1979 - P22

June 1, 2011
geometryperpendicular bisectorgeometry proposed

Problem Statement

Consider two quadrilaterals ABCDABCD and A′B′C′D′A'B'C'D' in an affine Euclidian plane such that AB=A′B′,BC=B′C′,CD=C′D′AB = A'B', BC = B'C', CD = C'D', and DA=D′A′DA = D'A'. Prove that the following two statements are true:
(a) If the diagonals BDBD and ACAC are mutually perpendicular, then the diagonals B′D′B'D' and A′C′A'C' are also mutually perpendicular.
(b) If the perpendicular bisector of BDBD intersects ACAC at MM, and that of B′D′B'D' intersects A′C′A'C' at M′M', then MA‾MC‾=M′A′‾M′C′‾\frac{\overline{MA}}{\overline{MC}}=\frac{\overline{M'A'}}{\overline{M'C'}} (if MC=0MC = 0 then M′C′=0M'C' = 0).
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