A French mathematician named Vecten extensively in 1817 investigated several interesting geometric properties of three squares constructed on the sides of a triangle; hence this construction is generally known as a Vecten configuration. This configuration is also known as the general Bride's Chair configuration as it generalizes Euclid's famous 'Bride's Chair' diagram for his proof of the Pythagorean theorem - see for example, Pellegrinetti & De Villiers (2021). Below are presented some lesser known variations of the Vecten (general Bride's Chair) configuration.
Variation 1 Conjecture
Given a Vecten (general Bride's Chair) configuration with squares, and respective centres D, E and F, respectively constructed outwards (or inwards) on the sides AB, BC and CA of △ABC.
1) If G, H and I are the midpoints of the segments as shown in the dynamic figure below, drag any of A, B, or C to change the configuration. Ensure that you also drag any of the vertices across a side opposite to it so that the constructed squares lie to the interior of the triangle. What do you notice?
2) Click on the 'Show other perpendiculars' button and repeat Step 1) above. What do you notice? Can you formulate a conjecture?
Vecten Variations
Challenge 1
3) Can you explain why (prove that) HE ⊥ GI and HE = GI as observed above (as well as the perpendicularity and equality of the other two pairs of corresponding segments, and also the concurrency of the three lines in P)?
Hint: Try using some of the Vecten line properties together with the Finsler–Hadwiger theorem (Detemple & Harold, 1996)1.
4) Can you prove the results in more than one way?
Footnote 1: The Finsler-Hadwiger theorem is a special case of a Fundamental Theorem of Similarity.
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Van Aubel Generalization & its Specialization to squares
Click on the 'Link to Van Aubel' button to navigate to a new sketch, which shows a specialization to squares, of a generalization of Van Aubel’s theorem involving similar rectangles erected on the sides of a quadrilateral (De Villers, 1998 & Silvester, 2006).
In this new figure, we now have FE = LD, FE ⊥ LD, KI = HJ, KI ⊥ HJ. Also note that these four lines are concurrent in O. Moreover, lines FE and LD are the angle bisectors of the right angle formed at O by the other two lines KI and HJ; and therefore these four lines are at 45o to each other.
5) Drag any of A, B, C or D. Ensure that you also drag the quadrilateral into a concave as well as crossed case. What do you notice?
Click on the 'Show Pellegrinetti circle' button to show a circle which passes through O and the midpoints Y, X, P and Q, respectively of FE, LD, HJ and KI (Pellegrinetti, 2019; Pellegrinetti & De Villiers, 2022; De Villiers, 2023).
6) Now drag points A1 and A2 towards each other until they more or less coincide while carefully observing what happens to all the points above. What do you notice when these points more or less coincide?
Vecten Variation 2: Van Aubel Triangle Specialization
Click on the 'Link to Van Aubel △ Special' button to navigate to a new sketch which shows A1 and A2 from the previous sketch merged together to form △ABC showing a specialization to a triangle of the Van Aubel results above.
7) Drag any of A, B, or C to change the configuration. Ensure that you also drag any of the vertices across a side opposite to it so that the constructed squares lie to the interior of the triangle. What do you notice?
a) Click on the 'Show other line segments' button. What do you notice about the three line segments GB2, HJ and AB, etc.?
b) Click on the 'Show Pellegrinetti circle' button to show a circle which passes through O and the midpoints Y, X, MAB and MAC, respectively of FE, AD, HJ and KI.
8) Challenge: Can you explain why (prove that) your observations above for the triangle specialization is true?
Vecten Variations 1 & 2 Combined
9) Click on the 'Link to Van Aubel △ Special + Vecten Variation' button above to navigate to a new sketch showing of the 1st Vecten variation combined with the Van Aubel Triangle Specialization.
10) In this new sketch, it is easy to to see that Y, the intersection of GH and FI, and Z, the intersection of GI and HE, are concyclic with O, and lie on the circle with diameter HI. (Similarly, for the other perpendicular intersections and sides).
11) Drag any of A, B, or C to explore.
Submitted Paper
A joint paper 'Variations of Vecten configurations' by Hans Humenberger & myself about the above results has been submitted for publication to The Mathematical Gazette - all rights reserved.
Some Other Vecten Variations
12) If instead of constructing squares on the sides of the base triangle, we construct equilateral triangles we obtain Napoleon's Theorem and several interesting generalizations.
13) If equilateral triangles are constructed on the sides we also obtain the Fermat-Torricelli point as a point of concurrency, which has several different generalizations as well.
14) The areas of the three triangular 'flanks' formed between two adjacent squares on the sides of the base triangle are equal to each other as well as to the area of the base triangle. Though this result was apparently proven by Vecten, it was rediscovered by a schoolboy named Cross around 2004 (De Villiers, 2007).
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. (1998). Dual Generalizations of Van Aubel's theorem. The Mathematical Gazette, Vol. 82, No. 495, pp. 405-412.
Silvester, J.R. (2006). Extensions of a Theorem of Van Aubel. The Mathematical Gazette, Vol. 90, No. 517, pp. 2-12.
De Villiers, M. (2007). An example of the discovery function of proof - aka Vecten flank area result. Mathematics in School, Vol. 36, No. 4, pp. 9-11.
Pellegrinetti, D. (2019). The six-point circle for the quadrangle. International Journal of Geometry, Vol. 8, No. 2, pp. 5 - 13.
Lord, N.; Rigby, J. & Quadling, D. (2010). (The Bride's Chair)2. The Mathematical Gazette, Vol. 94, No. 530 (July 2010), pp. 239-246.
Pellegrinetti, D. & De Villiers, M. (2021). Forgotten Properties of the Van Aubel & Bride's Chair configurations. International Journal of Geometry, Vol. 10, No. 3, pp. 5 - 10.
Pellegrinetti, D. & De Villiers, M. (2022). An extension of the six-point circle theorem for a generalised Van Aubel configuration. The Mathematical Gazette, Vol. 106, No. 567, pp. 400 - 407.
De Villiers, M. (2023). An associated result of the Van Aubel configuration and its generalization. International Journal of Mathematical Education in Science and Technology, 54:3, pp. 462-472. (See Theorem 4).
Related Links
Finsler–Hadwiger theorem plus Gamow-Bottema's Invariant point
A Fundamental Theorem of Similarity
Van Aubel's Theorem and some Generalizations
Twin Circles for a Van Aubel configuration involving Similar Parallelograms
An associated result of the Van Aubel configuration and some generalizations
Some Corollaries to Van Aubel Generalizations
A Vecten area variation (Cross's theorem) & generalizations to quadrilaterals
Fermat-Torricelli Point Generalization
Napoleon's Theorem: Generalizations & Converses
Some External Links
Bride's Chair
Vecten points
Finsler–Hadwiger theorem
Outer Vecten Triangle
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Created by Michael de Villiers with WebSketchpad, 11 Nov 2023; updated 20/27 Nov 2023; 15 Feb 2024.