Product of Diagonals in Regular N-gon

  Let ε0 = 1, ..., εN-1 be the vertices of a regular N-gon inscribed on the unit circle. Show that the product of the lengths of all diagonals of the N-gon emanating from ε0 is N.

Solution

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Copyright © 1996-2017 Alexander Bogomolny

  Let ε0 = 1, ..., εN-1 be the vertices of a regular N-gon inscribed on the unit circle. Show that the product of the lengths of all diagonals of the N-gon emanating from ε0 is N.

Solution

Let's shift the N-gon one unit to the left so that the vertices will become ω = ε - 1. In particular, ω0 = ε0 - 1 = 0 falls onto the origin. We are thus interested in the product Πωk, where k ranges from 1 to N-1.

Vertices ε are the Nth roots of unity: εN = 1. Except for ε0, all other ε's are the roots of the equation

  εN-1 + εN-2 + ... + ε + 1 = 0.

This means that ω's satisfy the equation

  (ω + 1)N-1 + (ω + 1)N-2 + ... + (ω + 1) + 1 = 0.

This is a polynomial equation of degree N-1 with the free coefficient equal to N and the roots ω1, ω2, ..., ωN-1. As is well known, the product of all roots of a polynomial equation is given by its free coefficient (up to a sign), in particular

  Πωk = (-1)N-1N,

which shows that

  Π|ωk| = |Πωk| = N.

Exactly what we set out to prove.

Complex Numbers

  1. Algebraic Structure of Complex Numbers
  2. Division of Complex Numbers
  3. Useful Identities Among Complex Numbers
  4. Useful Inequalities Among Complex Numbers
  5. Trigonometric Form of Complex Numbers
  6. Real and Complex Products of Complex Numbers
  7. Complex Numbers and Geometry
  8. Plane Isometries As Complex Functions
  9. Remarks on the History of Complex Numbers
  10. Complex Numbers: an Interactive Gizmo
  11. Cartesian Coordinate System
  12. Fundamental Theorem of Algebra
  13. Complex Number To a Complex Power May Be Real
  14. One can't compare two complex numbers
  15. Riemann Sphere and Möbius Transformation
  16. Problems

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Copyright © 1996-2017 Alexander Bogomolny

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