Dào Thanh Oai (2016, p. 19) posed the following interesting perpendicularity generalization of Van Aubel's quadrilateral theorem, which is distinctly different from the similar rectangles generalization given in De Villiers (1998) and Silvester (2006).
Dào Thanh Oai's Perpendicular Lines Van Aubel Theorem
Given any quadrilateral ABCD with triangles ABP, BCQ, CDR and DAS constructed on the sides so that ∠PAB = ∠SAD, ∠PBA = ∠QBC, ∠QCB = ∠RCD and ∠RDC = ∠SDA, and ∠PBA + ∠SDA = 90° and ∠SAD + ∠RCD = 90°, then PR ⟂ QS.
Investigate
Use the dynamic sketch below and investigate further by dragging.
1) Does the result still hold when the quadrilateral ABCD becomes concave or crossed?
2) Does the result still hold when the triangle lie 'inwards'?
Challenge
3) Can you explain why (prove that) the result is true for all the cases you observed above?
4) Can you prove it in more than one way?
Note: While the general result is fairly straight forward to prove using the methods of complex numbers (see Nicollier Proofs), the challenge remains to find purely geometry proofs.
Dào Thanh Oai's Perpendicular Lines Van Aubel Generalization
References
De Villiers, M. (1998). Dual Generalizations of Van Aubel's Theorem. Mathematical Gazette, Vol. 82, November, pp. 405-412.
Silvester, J. (2002). Extensions of a theorem of Van Aubel. Mathematical Gazette, Vol. 90, No. 517 (March), pp. 2-12.
Oai, D. O. (2016). Generalizations of some famous classical Euclidean geometry theorems. International Journal of Computer Discovered Mathematics, Volume 1, No.3, pp. 13-20. (See Problem 12 on p. 19).
Special Cases
4) How does the above theorem compare to the similar rectangles arrangement, also giving perpendicular lines, as discussed in De Villiers (1998) and Silvester (2006)?
(Specifically have a look at Van Aubel's Theorem and some Generalizations and click on the 'Link to Similar Rectangles' button).
5) To check your observation in 4) above, click on the 'Link to Dao Van Aubel Special 1' button.
In this special case, the point P has been merged to the perpendicular bisector of AB. What do you notice?
6) As can clearly be seen, we now have in this case, two similar right triangles BCQ and ADS on one pair of opposite sides, and on the other pair of opposite sides, we have two isosceles triangles with the apex angles supplementary. (This means on sides AB and CD we can construct similar rectangles (though one is 'rotated' in respect of the other) while on the other two sides we can construct similar rhombi. But clearly this is not the same configuration as the one in De Villiers (1998) and Silvester (2006)).
7) Lastly, click on the 'Link to Dao Van Aubel Special 2' button to view and manipulate the situation when ∠PAB in the dynamic sketch in 5) becomes 45°, and Dào's theorem becomes Van Aubel's theorem.
Related Links
A Hierarchical Classification of Quadrilaterals
Definitions and some Properties of Quadrilaterals
Van Aubel's Theorem and some Generalizations
A Van Aubel like property of an Equidiagonal Quadrilateral
A Van Aubel like property of an Orthodiagonal Quadrilateral
Some Corollaries to Van Aubel Generalizations
Klingens' theorem of two intersecting circles or two adjacent isosceles triangles
The Lux Problem
Quadrilateral Balancing Theorem: Another 'Van Aubel-like' theorem
More Properties of a Bisect-diagonal Quadrilateral
An associated result of the Van Aubel configuration and some generalizations
The Vertex Centroid of a Van Aubel Result involving Similar Quadrilaterals and its Further Generalization
Twin Circles for a Van Aubel configuration involving Similar Parallelograms
A Fundamental Theorem of Similarity
Finsler-Hadwiger Theorem plus Gamow-Bottema's Invariant Point
Pompe's Hexagon Theorem
Sum of Two Rotations Theorem
SA Mathematics Olympiad Problem 2016, Round 1, Question 20
SA Mathematics Olympiad 2022, Round 2, Q25
An extension of the IMO 2014 Problem 4
A 1999 British Mathematics Olympiad Problem and its dual
Dirk Laurie Tribute Problem
Extangential Quadrilateral
Triangulated Tangential Hexagon theorem
Theorem of Gusić & Mladinić
Conway's Circle Theorem as special case of Side Divider (Windscreen Wiper) Theorem
Japanese Circumscribed Quadrilateral Theorem
External Links
Equidiagonal quadrilateral (Wikipedia)
Orthodiagonal quadrilateral (Wikipedia)
Dào Thanh Oai's (perpendicular lines) Generalization of Van Aubel's Theorem (Cut the Knot)
SA Mathematics Olympiad
Questions and worked solutions for past South African Mathematics Olympiad papers can be found at this link.
(Note, however, that prospective users will need to register and log in to be able to view past papers and solutions.)
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