The following theorem is named after Finsler & Hadwiger (1937), but the result may be older than that.
Finsler-Hadwiger Theorem
Given two squares ABCD and AEFG sharing a vertex A as shown below. Then the respective midpoints, H and J, of the segments DE and BG, together with the centres I and K of the squares, form another square (Detemple & Harold, 1996).
Additional Properties
1) Click on the 'Show lines DG & BE' button. What do you notice?
2) Click on the 'Show lines CF & AL' button. What do you notice?
3) Can you formulate conjectures? Check your conjectures by dragging.
Finsler-Hadwiger Theorem plus Gamow-Bottema's Invariant Point
Comments
a) The Finsler-Hadwiger theorem is a special case of a Fundamental Theorem of Similarity.
b) The additional properties in 1)-3) above were already studied & proved by Vecten in 1817 in relation to three squares constructed on the sides of a triangle; so these results are directly related to the so-called Vecten points & lines.
Challenge 1
4) Can you explain why (prove that) the Finsler-Hadwiger theorem is true as well as your other conjectures above?
5) Can you prove the results in more than one way?
..........
Gamow-Bottema's Invariant Point
I've taken the liberty here of calling the fixed point H above, the Gamow-Bottema point, in reference to Gamow (1947) who provided a proof as well as to Bottema who apparently proved it independently (van Lamoen, 2001).
Click on the 'Link to Gamow-Bottema' button to navigate to a new sketch, which shows a slightly different dynamic version of the same configuration.
6) Drag A anywhere in the sketch. What do you notice about H?
7) Drag B or G, then repeat step 6 above.
Challenge 2
8) Can you explain why (prove that) the Gamow-Bottema point H is fixed and why its position remains invariant if B and G remain fixed?
9) Can you prove the result in more than one way?
Hint: Try using complex numbers, vectors or the composition (sum) of two rotations.
Further Generalization
10) Can you generalize the Finsler-Hadwiger and Gamow-Bottema theorems further by considering similar rectangles, similar rhombi, similar parallelograms, regular polygons, etc.?
11) Compare your answers to the above with the equivalent result to Gamow-Bottema (and a further generalization) at the intriguing Pirate Treasure Hunt problem (Gamow, 1947; De Villiers, 2016).
12) The Finsler-Hadwiger theorem and its additional properties can also be applied to Van Aubel's theorem for a quadrilateral & some generalizations.
References
Detemple, D. & Harold, S. (1996). A round-up of square problems. Mathematics Magazine, 69(1), pp. 15–27 (See problem 8, pp. 20–21 & problem 15, pp. 25–26).
De Villiers, M. (2016). An explanatory, transformation geometry proof of a classic treasure-hunt problem and its generalization. International Journal of Mathematical Education in Science and Technology, DOI: 10.1080/0020739X.2016.1210245.
(Note: The IJMEST paper above was based on talks given at the Annual Congress of the Association of Mathematics Education of South Africa (AMESA), 3–7 July 2000, University of Free State, Bloemfontein,
South Africa, as well as at the Annual Congress of the Mathematical Association (MA), 13–16
April 2004, University of York, United Kingdom.)
Finsler, P. & Hadwiger, H. (1937). Einige Relationen im Dreieck, Comment. Helv. 10, pp. 316-326.
Gamow, G. (1947). One, Two, Three ... Infinity (12.5 MB). Dover Publications, New York, pp. 35-37.
van Lamoen, F. (2001). Friendship Among Triangle Centers. Forum Geometricorum, Vol 1, pp. 1–6.
Related Links
A Fundamental Theorem of Similarity
Pirate Treasure Hunt and a Generalization
Sum of Two Rotations Theorem
Van Aubel's Theorem and some Generalizations
Pompe's Hexagon Theorem
Some Variations of Vecten configurations
Fermat-Torricelli Point Generalization
Napoleon's Theorem: Generalizations & Converses
Some External Links
Bottema's Theorem
Vecten points
Finsler–Hadwiger theorem
Outer Vecten Triangle
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Created by Michael de Villiers with WebSketchpad, 20 Nov 2023.