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?

**Background**:

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.