Finsler-Hadwiger Theorem

Let the Squares $\vbox{\hrule height.6pt\hbox{\vrule width.6pt height6pt \kern6.4pt \vrule width.6pt} \hrule height.6pt}ABCD$ and $\vbox{\hrule height.6pt\hbox{\vrule width.6pt height6pt \kern6.4pt \vrule width.6pt} \hrule height.6pt}AB'C'D'$ share a common Vertex $A$. The midpoints $Q$ and $S$ of the segments $B'D$ and $BD'$ together with the centers of the original squares $R$ and $T$ then form another square $\vbox{\hrule height.6pt\hbox{\vrule width.6pt height6pt \kern6.4pt \vrule width.6pt} \hrule height.6pt}QRST$. This theorem is a special case of the Fundamental Theorem of Directly Similar Figures (Detemple and Harold 1996).

See also Fundamental Theorem of Directly Similar Figures, Square

References

Detemple, D. and Harold, S. ``A Round-Up of Square Problems.'' Math. Mag. 69, 15-27, 1996.

Finsler, P. and Hadwiger, H. ``Einige Relationen im Dreieck.'' Comment. Helv. 10, 316-326, 1937.

Fisher, J. C.; Ruoff, D.; and Shileto, J. ``Polygons and Polynomials.'' In The Geometric Vein: The Coxeter Festschrift. New York: Springer-Verlag, 321-333, 1981.