Definition of the construction
Suppose that the vertices of the quadrilateral are given by . Let be the perpendicular bisectors of sides respectively. Then their intersections , with subscripts considered modulo 4, form the consequent quadrilateral . The construction is then iterated on to produce and so on. An equivalent construction can be obtained by letting the vertices of be the circumcenters of the 4 triangles formed by selecting combinations of 3 vertices of .Properties
1. If is not cyclic, then is not degenerate.J. King, Quadrilaterals formed by perpendicular bisectors, in ''Geometry Turned On'', (ed. J. King), MAA Notes 41, 1997, pp. 29–32. 2. Quadrilateral is never cyclic. Combining #1 and #2, is always nondegenrate. 3. Quadrilaterals and are homothetic, and in particular, similar.G. C. Shephard, The perpendicular bisector construction, ''Geom. Dedicata'', 56 (1995) 75–84. Quadrilaterals and are also homothetic. 3. The perpendicular bisector construction can be reversed viaReferences
* J. Langr, Problem E1050, ''Amer. Math. Monthly'', 60 (1953) 551. * V. V. Prasolov, ''Plane Geometry Problems'', vol. 1 (in Russian), 1991; Problem 6.31. * V. V. Prasolov, ''Problems in Plane and Solid Geometry'', vol. 1 (translated by D. Leites), available at http://students.imsa.edu/~tliu/math/planegeo.eps{{Dead link, date=May 2020 , bot=InternetArchiveBot , fix-attempted=yes . * D. Bennett, Dynamic geometry renews interest in an old problem, in ''Geometry Turned On'', (ed. J. King), MAA Notes 41, 1997, pp. 25–28. * J. King, Quadrilaterals formed by perpendicular bisectors, in ''Geometry Turned On'', (ed. J. King), MAA Notes 41, 1997, pp. 29–32. * G. C. Shephard, The perpendicular bisector construction, ''Geom. Dedicata'', 56 (1995) 75–84. * A. Bogomolny, Quadrilaterals formed by perpendicular bisectors, ''Interactive Mathematics Miscellany and Puzzles'', http://www.cut-the-knot.org/Curriculum/Geometry/PerpBisectQuadri.shtml. * B. Grünbaum, On quadrangles derived from quadrangles—Part 3, ''Geombinatorics'' 7(1998), 88–94. * O. Radko and E. Tsukerman, The Perpendicular Bisector Construction, the Isoptic Point and the Simson Line of a Quadrilateral, ''Forum Geometricorum'' 12: 161–189 (2012). Quadrilaterals