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Perpendicular bisector construction of a quadrilateral

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 constructionEdit

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.

 
First iteration of the perpendicular bisector construction

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  .

PropertiesEdit

1. If   is not cyclic, then   is not degenerate.[1]

2. Quadrilateral   is never cyclic.[1] Combining #1 and #2,   is always nondegenrate.

3. Quadrilaterals   and   are homothetic, and in particular, similar.[2] Quadrilaterals   and   are also homothetic.

3. The perpendicular bisector construction can be reversed via isogonal conjugation.[3] 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[3]

 

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.[3]

ReferencesEdit

  1. ^ a b J. King, Quadrilaterals formed by perpendicular bisectors, in Geometry Turned On, (ed. J. King), MAA Notes 41, 1997, pp. 29–32.
  2. ^ G. C. Shephard, The perpendicular bisector construction, Geom. Dedicata, 56 (1995) 75–84.
  3. ^ a b c 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).
  • 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.
  • 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).