In geometry, the perpendicular bisector construction of a quadrilateral is a construction which produces a new quadrilateral from a given quadrilateral using the perpendicular bisectors to the sides of the former quadrilateral. This construction arises naturally in an attempt to find a replacement for the circumcenter of a quadrilateral in the case that is non-cyclic.

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 via isogonal conjugation.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). That is, given , it is possible to construct .

4. Let be the angles of . For every , the ratio of areas of and is given by

5. If is convex then the sequence of quadrilaterals converges to the isoptic point of , which is also the isoptic point for every . Similarly, if is concave, then the sequence obtained by reversing the construction converges to the Isoptic Point of the 's.

6. If is tangential then is also tangential..

References

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) 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).

External links Perpendicular-Bisectors of Circumscribed Quadrilateral Theorem at Dynamic Geometry Sketches, interactive dynamic geometry sketches.