Circumscribed Quadrilateral Perpendicular-Bisectors Theorem

Perpendicular Bisectors (or Circumcentres) of Circumscribed Quadrilateral Theorem

The following theorem was experimentally discovered by me in 1991 or 1992 using dynamic geometry. A proof by Jordan Tabov from Bulgaria was given in the 1st draft edition in 1994 of my book Some Adventures in Euclidean Geometry (free to download). More background details are given further down.

Perpendicular Bisectors of Tangential 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.
Note: The dynamic sketch below also illustrates Pitot's theorem, and the necessary and sufficient condition for a quadrilateral to have an incircle, namely, that the two sums of its opposite sides are equal.

Perpendicular Bisectors (or Circumcentres) of Circumscribed Quadrilateral Theorem

Exploring Different Cases
As shown in a systematic analysis in a paper by Humenberger (2023), there are actually 6 different types of quadrilaterals formed when 4 tangent lines are drawn to a circle. The convex tangential quadrilateral ABCD shown with an incircle in the sketch above is obviously one example we can get from 4 lines tangential to a circle, but we can also obtain a concave tangential quadrilateral (with an incircle), or a convex, a concave or a crossed (two types) extangential quadrilateral (with an excircle) - 6 different types in total.
So it raises the natural question: do the perpendicular bisectors of the sides of the other 5 types of cases also produce another tangential quadrilateral (with an incircle) or an extangential quadrilateral (with an excircle)?
Further investigation confirms that it indeed does so in each case. For the cases illustrated below, note that for a quadrilateral to be extangential, i.e. have an excircle, it is a necessary and sufficient condition that two (distinct) sums of adjacent sides are equal.
To dynamically explore each of the other 5 cases for ABCD, navigate to the relevant pages using the 'Link to' buttons as follows:
i) Click on the 'Link to ABCD crossed excircle1' button to navigate to the 1st case where ABCD is crossed with an excircle.
ii) Continue clicking the other green buttons to view the other 4 cases until you reach the last case where ABCD is again crossed with an excircle (but differently from the 1st case).

But there are also 3 different ways in which a quadrilateral EFGH can be formed by the four perpendicular bisectors of ABCD, and where the (extended) sides of EFGH are tangent to a circle. So in total there are 18 different cases.
iii) For example, click on the 'Link to EFGH concave incircle' button to navigate to the case where ABCD is convex with an incircle and where EFGH is concave with an incircle.
iv) Lastly, click on the 'Link to EFGH crossed excircle' button to navigate to the case where ABCD is convex with an incircle and where EFGH is crossed with an excircle.
v) Readers may wish to explore the other remaining cases with constructions of their own.

Generalized Theorem
The original theorem can therefore now be more generally formulated: The perpendicular bisectors of the sides of a quadrilateral with an incircle or an excircle form another quadrilateral with an incircle or an excircle.

Challenge
Can you prove the result? Can you prove it in more than one way? Can you prove it purely geometrically? Is your proof general enough to cover all the cases above?

Background Details
1) This interesting result was experimentally discovered by myself 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 book Some Adventures in Euclidean Geometry (free to download), first published in 1994. An abridged version of Tabov's proof is available: here.
2) A little later after my discovery, round about 1993/94, I also wrote to the well-known geometer HSM Coxeter (1907-2003) at the University of Toronto about this result and 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 geometric proof using inversion: here.
5) In 2023, Hans Humenberger found that 6 different types of quadrilaterals are formed by four lines tangent to a circle in his paper Unusual Cyclic and Tangential Quadrilaterals – An Overview. This paper prompted the investigation of the different cases of the 'perpendicular bisectors of the sides' theorem discussed above.

Related Properties
6) In 2006, Alexei Myakishev in his article On Two Remarkable Lines Related to a Quadrilateral, Forum Geom., 6, 289–295 proved in general, that the quadrilateral formed by the circumcentres of the subdividing triangles of any quadrilateral is affine equivalent to the original. Similarly, but independently, Maria Mammana & Biagio Micale in their 2008 paper Quadrilaterals of triangle centres in the Mathematical Gazette, Vol. 92, No. 525 (November), pp. 466-475 (see Theorem 4), also proved this affine relationship. So for the original result above, this implies that the two tangential quadrilaterals ABCD and EFGH are affine equivalent, but note their affine equivalence is not sufficient to conclude that since it is given that ABCD is tangential, this implies that EFGH be must also be tangential. For example, a square and a rectangle are affine equivalent, but a square is tangential while a rectangle is not.
7) Another interesting property of the configuration shown above is that the angle bisectors of EFGH are respectively parallel to the angle bisectors of ABCD. This is easy to prove & left to the reader.
8) In an article published by Branko Grünbaum (1929-2018) in 1993, he used Mathematica to show, 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 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.
9) A generalization of Grünbaum's theorem mentioned in 8) above, to any (non-cyclic) quadrilateral, and that Q₂ is homothetic to Q, was proved purely geometrically by Maria Mammana & Biagio Micale in their 2008 paper Quadrilaterals of triangle centres in the Mathematical Gazette, Vol. 92, No. 525 (November), pp. 466-475 (see Theorem 5). Morever, as pointed out to me in a personal communication (2022) from Vladimir Dubrovsky, Kolmogorov School of Moscow State University, it's not hard to see that the ratio of the homothety equals -(cot A + cot C)(cot B + cot D)/4.

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

External Links
Tangential quadrilateral (Wikipedia)
Tangential Quadrilateral (Wolfram MathWorld)
A Problem in a Special Tangential Quadrilateral (Cut The Knot)
Ex-tangential quadrilateral (Wikipedia)
Perpendicular bisector construction of a 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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Michael de Villiers, created 19 Dec 2009; updated to WebSketchpad, 22 March 2021; 14; 18 July 2022; 12 April 2025.