# Squares on Sides of a Quadrilateral

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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

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.

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.

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.

The problem has a simple consequence.

There is an additional exploration of this configuration.

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.)There is an interesting degenerate case.

### References

- E. J. Barbeau, M. S. Klamkin, W. O. J. Moser,
*Five Hundred Mathematical Challenges*, MAA, 1995, #320b - 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 - R. L. Finney,
__Dynamic Proofs of Euclidean Theorems__,*Math Magazine*, 43, pp. 177-185. - M. Gardner,
*Mathematical Circus*, Vintage, 1981 - I. M. Yaglom,
*Geometric Transformations I*, MAA, 1962

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