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
[edit]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.
![](http://upload.wikimedia.org/wikipedia/commons/thumb/0/04/PerpendicularBisectorConstruction.svg/360px-PerpendicularBisectorConstruction.svg.png)
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
[edit]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]
6. If is tangential then is also tangential.[4]
References
[edit]- ^ a b 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 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).
- ^ de Villiers, Michael (2009), Some Adventures in Euclidean Geometry, Dynamic Mathematics Learning, p. 192-193, ISBN 9780557102952.
- 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
[edit]![](http://upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png)
- Perpendicular-Bisectors of Circumscribed Quadrilateral Theorem at Dynamic Geometry Sketches, interactive dynamic geometry sketches.