<> Circumscribed Quadrilateral Perpendicular-Bisectors Theorem

Perpendicular-Bisectors (or Circumcentres) of Circumscribed Quadrilateral Theorem

The perpendicular bisectors of the sides of a quadrilateral circumscribed around a circle (a tangential quad) form another quadrilateral circumscribed around a circle (another tangential quad). Alternatively, but equivalently formulated, the circumcentres of triangles ABC, BCD, CDA and DAB of a quadrilateral ABCD circumscribed around a circle form another quadrilateral circumscribed around a circle.

Perpendicular-Bisectors (or Circumcentres) of Circumscribed Quadrilateral Theorem

Challenge: Can you prove the result?

1) This interesting result was experimentally discovered by the author about 1991/92 using dynamic geometry, and so far have not yet seen/found earlier references to this result. Unable at the time to prove the result myself, I wrote to Jordan Tabov from the Bulgarian Academy of Sciences, who produced a neat trigonometric proof of the result, and which appears on p. 192-193 of my Some Adventures in Euclidean Geometry book, first published in 1994. An abridged version of Tabov's proof is available: here.
2) A little later, round about 1993/94, I also wrote to the well-known geometer HSM Coxeter (1907-2003) at the University of Toronto about this result & he confirmed that he'd not seen it before. A colleague of his, John Wilker, then produced a proof similar to that of Tabov (link given above).
3) A few years later, round about 2004/2005, it was apparently independently rediscovered by Marcello Tarquini, and then proved by Darij Grinberg. A copy of Grinberg's paper involving his synthetic proof is available: here.
4) More recently, Stanislaw Hauke found the problem listed as unsolved Problem 1351 at Antonio Gutierrez's site: GoGeometry, and in March 2018, he produced the following neat geometric proof: here.

More Properties:
1) Another interesting property of the configuration shown above is that the angle bisectors of EFGH are respectively parallel to the angle bisectors of CBAD. This is easy to prove & left to the reader.
2) In an article published by Branko Grünbaum (1929-2018) in 1993, he showed using Mathematica, that in general, if the perpendicular bisectors of any quadrilateral Q form another quadrilateral Q₁, and the perpendicular bisectors of Q₁ are also constructed to form quadrilateral Q₂, then Q₂ is similar to Q. This remarkable theorem implies that for the above dynamic configuration that the perpendicular bisectors of EFGH form another circumscribed (tangential) quadrilateral similar to the original.
Reference: Grünbaum, B. (1993). Quadrangles, Pentagons and Computers. Geombinatorics, 3, pp. 4-9.

Read & explore more about the properties of circumscribed (tangential) quadrilaterals at this interactive webpage.


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Michael de Villiers, created 19 Dec 2009; updated 22 March 2021.