Minkowski's second theorem: Difference between revisions
Deltahedron (talk | contribs) m Deltahedron moved page User:Deltahedron/Minkowski's second theorem to Minkowski's second theorem: Publish |
→Statement of the theorem: Replaced the Rs with \mathbb{R}s. |
||
Line 1: | Line 1: | ||
In [[mathematics]], the '''perpendicular bisector construction''' 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 very naturally and has been considered by many authors over the years. | |||
==Definition of the construction== | |||
Suppose that the [[vertex (geometry)|vertices]] of the [[quadrilateral]] <math> Q </math> are given by <math> Q_1,Q_2,Q_3,Q_4 </math>. Let <math> b_1,b_2,b_3,b_4 </math> be the perpendicular bisectors of sides <math> Q_1Q_2,Q_2Q_3,Q_3Q_4,Q_4Q_1 </math> respectively. Then their intersections <math> Q_i^{(2)}=b_{i+2}b_{i+3} </math>, with subscripts considered modulo 4, form the consequent quadrilateral <math> Q^{(2)} </math>. The construction is then iterated on <math> Q^{(2)} </math> to produce <math> Q^{(3)} </math> and so on. | |||
[[File:PerpendicularBisectorConstruction.svg|thumb|360px|First iteration of the perpendicular bisector construction]] | |||
An equivalent construction can be obtained by letting the vertices of <math> Q^{(i+1)} </math> be the [[circumcenter]]s of the 4 triangles formed by selecting combinations of 3 vertices of <math> Q^{(i)} </math>. | |||
==Properties== | |||
1. If <math> Q^{(1)} </math> is not cyclic, then <math> Q^{(2)} </math> is not degenerate.<ref name=King>J. King, Quadrilaterals formed by perpendicular bisectors, in ''Geometry Turned On'', (ed. J. King), MAA Notes 41, 1997, pp. 29–32.</ref> | |||
2. Quadrilateral <math> Q^{(2)} </math> is never cyclic.<ref name="King" /> | |||
3. Quadrilaterals <math> Q^{(1)} </math> and <math> Q^{(3)} </math> are [[Homothety|homothetic]], and in particular, [[Similarity (geometry)|similar]].<ref name=Shephard>G. C. Shephard, The perpendicular bisector construction, ''Geom. Dedicata'', 56 (1995) 75–84.</ref> Quadrilaterals <math> Q^{(2)} </math> and <math> Q^{(4)} </math> are also homothetic. | |||
3. The Perpendicular bisector construction can be reversed via [[isogonal conjugation]].<ref name=RadkoTsukerman>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).</ref> That is, given <math> Q^{(i+1)} </math>, it is possible to construct <math> Q^{(i)} </math>. | |||
4. Let <math> \alpha, \beta, \gamma, \delta </math> be the angles of <math> Q^{(1)} </math>. For every <math> i </math>, the ratio of areas of <math> Q^{(i)} </math> and <math> Q^{(i+1)} </math> is given by<ref name="RadkoTsukerman" /> | |||
<div style="text-align: center;"><math> (1/4)(cot(\alpha)+cot(\gamma))(cot(\beta)+cot(\delta)). </math></div> | |||
5. If <math> Q^{(1)} </math> is convex then the sequence of quadrilaterals <math> Q^{(1)}, Q^{(2)},\ldots </math> converges to the [[Isoptic Point]] of <math> Q^{(1)} </math>, which is also the Isoptic Point for every <math> Q^{(i)} </math>. Similarly, if <math> Q^{(1)} </math> is concave, then the sequence <math> Q^{(1)}, Q^{(0)}, Q^{(-1)},\ldots </math> obtained by reversing the construction converges to the Isoptic Point of the <math> Q^{(i)} </math>'s.<ref name="RadkoTsukerman" /> | |||
==Motivating discussion== | |||
The perpendicular bisector construction arises naturally in an attempt to find a replacement for the [[circumcenter]] of a quadrilateral in the case that it is noncyclic. | |||
==References== | |||
{{reflist}} | |||
<!--- After listing your sources please cite them using inline citations and place them after the information they cite. Please see http://en.wikipedia.org/wiki/Wikipedia:REFB for instructions on how to add citations. ---> | |||
* 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). | |||
[[Category:Quadrilaterals]] |
Revision as of 16:02, 30 August 2013
In mathematics, the perpendicular bisector construction 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 very naturally and has been considered by many authors over the years.
Definition of the construction
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.
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
1. If is not cyclic, then is not degenerate.[1]
2. Quadrilateral is never cyclic.[1]
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]
Motivating discussion
The perpendicular bisector construction arises naturally in an attempt to find a replacement for the circumcenter of a quadrilateral in the case that it is noncyclic.
References
43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.
- 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).
- ↑ 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.
- ↑ G. C. Shephard, The perpendicular bisector construction, Geom. Dedicata, 56 (1995) 75–84.
- ↑ 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).