A Problem of Hinged Squares
What is it?
A Mathematical Droodle

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Discussion

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Copyright © 1996-2015 Alexander Bogomolny

Discussion

In the applet two squares OABC and OPQR share a common vertex. M and N are centers of the two squares. K and L are the midpoints of AP and CR, respectively. It appears that the quadrilateral KMLN is a square. This is indeed so and is known as the Finsler-Hadwiger theorem.

Consider triangles AOR and COP. OR = OP. Moreover one may be obtained from the other by a suitable rotation through 90° around O. OA = OC. These too map on each other by the same rotations. Finally, the same is true of triangles AOR and COP, which are thus equal. From here, AR = CP and the two line segments are perpendicular.

In ΔAPR, KN that connects midpoints of sides AP and PR is parallel and equal to half AR. Similarly, in ΔACP, KM is parallel and equal to half CP. Therefore, KM = KN and the two are perpendicular.

In an absolutely similar fashion, from the geometry of triangles CPR and ACR, LM = LN and the two are perpendicular. This proves that KMLN is in fact a square.

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Here is another proof. Let M₉₀, N₉₀, and K₁₈₀ denote the rotations around M through 90°, around N through 90°, and around K through 180°. For example, M₉₀(A) = O, N₉₀(O) = P, K₁₈₀(P) = A. The composition of the three rotations has a fixed point at A and is a rotation through 360° = 0° (mod 360). It is therefore a translation with a fixed point, i.e. the identity transformation.

Denote the image of M under N₉₀ as M': M' = N₉₀(M). ( ΔMNM' is isosceles with angle N being 90°.) Then

M = K₁₈₀(N₉₀(M₉₀(M))) = K₁₈₀(N₉₀(M)) = K₁₈₀(M'),

It follows that M = K₁₈₀(M'). But K₁₈₀(M') is a reflection in K. Therefore K is the midpoint of MM'. Therefore ΔMNK is isosceles with angle K being 90°.

Remark

1. Squares OABC and OPQR are constructed on the sides OA and OP of ΔAOP. We have just showed that the center of the square built on the line MN of their centers coincides with the midpoint K of AP. This fact is known as Neuberg's Theorem.

2. The Finsler-Hadwiger theorem is also a special case of the Fundamental Theorem of Directly Similar Figures

3. The theorem is directly equivalent to one of Thébault's statements.

References

1. R. L. Finney, Dynamic Proofs of Euclidean Theorems, Math Magazine, 43, pp. 177-185.
2. R. Honsberger, In Pólya's Footsteps, MAA, 1997

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Copyright © 1996-2015 Alexander Bogomolny