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 ${\displaystyle Q}$ are given by ${\displaystyle Q_{1},Q_{2},Q_{3},Q_{4}}$. Let ${\displaystyle b_{1},b_{2},b_{3},b_{4}}$ be the perpendicular bisectors of sides ${\displaystyle Q_{1}Q_{2},Q_{2}Q_{3},Q_{3}Q_{4},Q_{4}Q_{1}}$ respectively. Then their intersections ${\displaystyle Q_{i}^{(2)}=b_{i+2}b_{i+3}}$, with subscripts considered modulo 4, form the consequent quadrilateral ${\displaystyle Q^{(2)}}$. The construction is then iterated on ${\displaystyle Q^{(2)}}$ to produce ${\displaystyle Q^{(3)}}$ and so on.

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

Properties

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

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

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

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

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

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

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

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

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^{[permanent 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).

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