Squares on Sides of a Quadrilateral
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A Mathematical Droodle

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Copyright © 1996-2018 Alexander Bogomolny

Discussion

Erect similarly oriented squares on the sides of a quadrilateral $ABCD.$ Denote their centers successively $O_{1},$ $O_{2},$ $O_{3},$ and $O_{4}.$ Then

(1)

$O_{1}O_{3} = O_{2}O_{4}$

and

(2)

$O_{1}O_{3} \perp O_{2}O_{4}.$

Proof

Let $M$ be the midpoint of $AC.$ By the Finsler-Hadwiger theorem,

$MO_{1} = MO_{2}$ and $MO_{1} \perp MO_{2}$

and also

$MO_{3} = MO_{4}$ and $MO_{3} \perp MO_{4}.$

Therefore, triangles $MO_{1}O_{3}$ and $MO_{2}O_{4}$ are equal and one is a rotation of the other through $90^{\circ}.$ Hence, (1) and (2) hold.

Remark

  1. If any two adjacent vertices of a quadrilateral coalesce into a point, the statement just proven becomes a property of the Bride's Chair configuration.

  2. The theorem we just proved is attributed to Van Aubel (Von Aubel in [Gardner, p. 176-178]) could also be found in [de Villiers, Yaglom, Finney] among others.

  3. The argument falls through in case one of $O_{1}O_{3}$ or $O_{2}O_{4}$ is of zero length. This case is considered elsewhere. There it is shown that the two distances are either both 0 or not. The former case occurs iff the diagonals of the quadrilateral are equal and perpendicular.

  4. The problem has a simple consequence.

  5. There is an additional exploration of this configuration.

  6. As a consequence, the quadrilateral with vertices at the midpoints of the sides of quadrilateral $O_{1}O_{2}O_{3}O_{4}$ is a square. (This was problem 11328, Am Math Monthly 114 December 2007.)

  7. There is an interesting degenerate case.

References

  1. E. J. Barbeau, M. S. Klamkin, W. O. J. Moser, Five Hundred Mathematical Challenges, MAA, 1995, #320b
  2. M. de Villiers, The Role of Proof in Investigative, Computer-based Geometry: Some Personal Reflections, in Geometry Turned On, MAA Notes 41, 1997, pp. 15-24
  3. R. L. Finney, Dynamic Proofs of Euclidean Theorems, Math Magazine, 43, pp. 177-185.
  4. M. Gardner, Mathematical Circus, Vintage, 1981
  5. I. M. Yaglom, Geometric Transformations I, MAA, 1962
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Copyright © 1996-2018 Alexander Bogomolny
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