The perpendicular bisectors of the sides of a quadrilateral circumscribed around a circle (a circum quad) form another quadrilateral circumscribed around a circle (circum 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 (circum quad).
Perpendicular-Bisectors (or Circumcentres) of Circumscribed Quadrilateral Theorem
Can you prove the result?
This interesting result was discovered by the author about 1992 using dynamic geometry, and so far it seems original. A few years later, it was independently rediscovered and proved by Darij Grinberg at Darij Grinberg Homepage.
A trigonometric proof of the result by Jordan Tabov from Bulgaria is given on p. 192-193 of Some Adventures in Euclidean Geometry, which is available for purchase as downloadable PDF or printed book at More Info.
Michael de Villiers, 19 Dec 2009.