In the Reader Investigations of the Amesa KZN Mathematics Journal, May 2005, Vol. 9, No. 1, I posed the following problem as problem no. 4 (which had been discovered in 2004 using the dynamic geometry software Sketchpad).
Problem
Given any orthodiagonal quadrilateral ABCD with AC ⟂ BD, and similar triangles AEB, CFB, CGD and AHD are constructed on the sides as shown below, prove that EG = FH.
Investigate
Use the dynamic sketch below and investigate further by dragging.
1) Does the result still hold when the orthodiagonal quadrilateral ABCD becomes concave or crossed?
2) Does the result still hold when the similar triangle lie 'inwards'?
Challenge
3) Can you explain why (prove that) the result is true for all the cases you observed above?
Van Aubel property of an Orthodiagonal Quadrilateral
Published Solution
Read the Solutions to Reader Investigations: May 2005 in AMESA KZN Mathematics Journal, December 2004, pp. 64-66, which gives two elegant proofs of this result (Problem 4) by Michael Fox from Leamington Spa, Warwickshire, UK.
Converse
Consider the following possible converse: Given any quadrilateral ABCD, and similar triangles AEB, CFB, CGD and AHD are constructed on the sides as shown above and EG = FH, then ABCD is orthodiagonal.
4) Is this converse true or not? If true, give a proof. If not, give a counter-example.
Further Generalization
5) What happens if ABCD is not an orthodiagonal quadrilateral? Can you generalize further to any quadrilateral?
6) Click on the 'Link to Similar Right Angles' button. What do you notice for an arbitrary quadrilateral ABCD?
7) Compare your observation in 5) above with the similar rhombi generalization at: Van Aubel's Theorem and some Generalizations (click on the 'Link to Similar Rhombi' button).
More Recent Updates
When the above problem 1-3 was posted at the Facebook Group Romantics of Geometry on 9 Oct 2024, the following responses were received:
1) Ichung Chen gave a short, general proof covering all cases using complex numbers. Go to: Ichung Proof Orthodiagonal Van Aubel.
2) Marian Cucoanes from Focsani, Romania proved the equidiagonal Van Aubel result using complex coordinates which covers all cases. Go to: Marion Proof Orthodiagonal Van Aubel.
3) Kousit Sett from India gave a link to the following related results for a quadrilateral with equal and perpendicular diagonals and which was earlier posted in the group on 7/06/2024. Go to: Marcio Perez Dedication
4) Other interesting dual special cases for quadrilaterals with equal and perpendicular diagonals were also earlier posted in 2020 in the group by Stanley Rabinowitz. Go to: Rabinowitz Special Cases.
5) On 17/18 Oct 2024, Jean-Pol Coulon from Tournai, Belgium, highlighted a nice duality with the following interesting related problem and proof on 17/18 October 2024 in the Romantics of Geometry Facebook Group in relation to an earlier post by Ercole Suppa, RG 13475, about a quadrilateral that is both orthodiagonal and equidiagonal. Go to: Jean-Pol Coulon Posting.
Related Links
A Hierarchical Classification of Quadrilaterals
Definitions and some Properties of Quadrilaterals
Van Aubel's Theorem and some Generalizations
Dào Thanh Oai's Perpendicular Lines Van Aubel Generalization
A Van Aubel like property of an Equidiagonal Quadrilateral
Quadrilateral Balancing Theorem: Another 'Van Aubel-like' theorem
More Properties of a Bisect-diagonal Quadrilateral
Some Corollaries to Van Aubel Generalizations
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
Klingens' theorem of two intersecting circles or two adjacent isosceles triangles
The Lux Problem
A Fundamental Theorem of Similarity
Finsler-Hadwiger Theorem plus Gamow-Bottema's Invariant Point
Pompe's Hexagon Theorem
Sum of Two Rotations Theorem
The Orthocentre Quadrilateral of a Quadrilateral
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
Opposite Side Quadrilateral Properties by Kalogerakis
Another Property of an Opposite Side Quadrilateral
Conway's Circle Theorem as special case of Side Divider (Windscreen Wiper) Theorem
Japanese Circumscribed Quadrilateral Theorem
External Links
Orthodiagonal quadrilateral (Wikipedia)
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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Created with WebSketchpad by Michael de Villiers, 9/10/26 Oct 2024.