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 construction

Suppose that the vertices of the quadrilateral $Q$ are given by $Q_1,Q_2,Q_3,Q_4$. Let $b_1,b_2,b_3,b_4$ be the perpendicular bisectors of sides $Q_1Q_2,Q_2Q_3,Q_3Q_4,Q_4Q_1$ respectively. Then their intersections $Q_i^=b_b_$, with subscripts considered modulo 4, form the consequent quadrilateral $Q^$. The construction is then iterated on $Q^$ to produce $Q^$ and so on.

An equivalent construction can be obtained by letting the vertices of $Q^$ be the circumcenters of the 4 triangles formed by selecting combinations of 3 vertices of $Q^$.

Properties

1. If $Q^$ is not cyclic, then $Q^$ 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 $Q^$ is never cyclic. Combining #1 and #2, $Q^$ is always nondegenrate.

3. Quadrilaterals $Q^$ and $Q^$ are homothetic, and in particular, similar.G. C. Shephard, The perpendicular bisector construction, ''Geom. Dedicata'', 56 (1995) 75–84. Quadrilaterals $Q^$ and $Q^$ 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 $Q^$, it is possible to construct $Q^$.

4. Let $\alpha, \beta, \gamma, \delta$ be the angles of $Q^$. For every $i$, the ratio of areas of $Q^$ and $Q^$ is given by

: $(1/4)(\cot(\alpha)+\cot(\gamma))(\cot(\beta)+\cot(\delta)).$

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

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), 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