Perpendicular bisector construction of a quadrilateral

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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 [math]\displaystyle{ Q }[/math] are given by [math]\displaystyle{ Q_1,Q_2,Q_3,Q_4 }[/math]. Let [math]\displaystyle{ b_1,b_2,b_3,b_4 }[/math] be the perpendicular bisectors of sides [math]\displaystyle{ Q_1Q_2,Q_2Q_3,Q_3Q_4,Q_4Q_1 }[/math] respectively. Then their intersections [math]\displaystyle{ Q_i^{(2)}=b_{i+2}b_{i+3} }[/math], with subscripts considered modulo 4, form the consequent quadrilateral [math]\displaystyle{ Q^{(2)} }[/math]. The construction is then iterated on [math]\displaystyle{ Q^{(2)} }[/math] to produce [math]\displaystyle{ Q^{(3)} }[/math] and so on.

First iteration of the perpendicular bisector construction

An equivalent construction can be obtained by letting the vertices of [math]\displaystyle{ Q^{(i+1)} }[/math] be the circumcenters of the 4 triangles formed by selecting combinations of 3 vertices of [math]\displaystyle{ Q^{(i)} }[/math].

Properties

1. If [math]\displaystyle{ Q^{(1)} }[/math] is not cyclic, then [math]\displaystyle{ Q^{(2)} }[/math] is not degenerate.[1]

2. Quadrilateral [math]\displaystyle{ Q^{(2)} }[/math] is never cyclic.[1] Combining #1 and #2, [math]\displaystyle{ Q^{(3)} }[/math] is always nondegenrate.

3. Quadrilaterals [math]\displaystyle{ Q^{(1)} }[/math] and [math]\displaystyle{ Q^{(3)} }[/math] are homothetic, and in particular, similar.[2] Quadrilaterals [math]\displaystyle{ Q^{(2)} }[/math] and [math]\displaystyle{ Q^{(4)} }[/math] are also homothetic.

3. The perpendicular bisector construction can be reversed via isogonal conjugation.[3] That is, given [math]\displaystyle{ Q^{(i+1)} }[/math], it is possible to construct [math]\displaystyle{ Q^{(i)} }[/math].

4. Let [math]\displaystyle{ \alpha, \beta, \gamma, \delta }[/math] be the angles of [math]\displaystyle{ Q^{(1)} }[/math]. For every [math]\displaystyle{ i }[/math], the ratio of areas of [math]\displaystyle{ Q^{(i)} }[/math] and [math]\displaystyle{ Q^{(i+1)} }[/math] is given by[3]

[math]\displaystyle{ (1/4)(\cot(\alpha)+\cot(\gamma))(\cot(\beta)+\cot(\delta)). }[/math]

5. If [math]\displaystyle{ Q^{(1)} }[/math] is convex then the sequence of quadrilaterals [math]\displaystyle{ Q^{(1)}, Q^{(2)},\ldots }[/math] converges to the isoptic point of [math]\displaystyle{ Q^{(1)} }[/math], which is also the isoptic point for every [math]\displaystyle{ Q^{(i)} }[/math]. Similarly, if [math]\displaystyle{ Q^{(1)} }[/math] is concave, then the sequence [math]\displaystyle{ Q^{(1)}, Q^{(0)}, Q^{(-1)},\ldots }[/math] obtained by reversing the construction converges to the Isoptic Point of the [math]\displaystyle{ Q^{(i)} }[/math]'s.[3]

6. If [math]\displaystyle{ Q^{(1)} }[/math] is tangential then [math]\displaystyle{ Q^{(2)} }[/math] is also tangential.[4]

References

  1. ↑ 1.0 1.1 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. ↑ 3.0 3.1 3.2 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).
  4. ↑ de Villiers, Michael (2009), Some Adventures in Euclidean Geometry, Dynamic Mathematics Learning, p. 192-193, ISBN 9780557102952, https://books.google.com/books?id=R7uCEqwsN40C&pg=PA58 .
  • 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[yes|permanent dead link|dead link}}].
  • 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



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