Rotations in Disguise
What Is This About?
A Mathematical Droodle

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Explanation

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Copyright © 1996-2018 Alexander Bogomolny

Rotations in Disguise

In one of O. Bottema's theorem, squares have been constructed on the sides of a triangle and a line joining two corners of the squares passed through the center of the square on the third side with the opposite orientation.

The applet below suggests an analogue of Bottema's theorem wherein equilateral triangles are constructed on the sides AC, BC of ΔABC outside the latter and another one on the side AB, with opposite orientation. The triangles are denoted AB'C, A'BC, and ABC'. If M' is the center of ΔABC', then A'M' = B'M' and angle A'M'B' = 120°.

This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at https://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.

What if applet does not run?

The assertion is an immediate consequence of a property of rotations. The product of two rotations is a rotation through the angle equal to the sum of the two rotation angles and the center can be easily obtained at the intersection of two rotation axes.

To apply this reasoning, consider two rotations through 60°: one centered at A, the other at B. B' first rotated around A into C which then continues around B into A'. The product is a rotation through 120° with the center at the center of ΔABC'.

Bottema's Theorem

  1. Bottema's Theorem
  2. An Elementary Proof of Bottema's Theorem
  3. Bottema's Theorem - Proof Without Words
  4. On Bottema's Shoulders
  5. On Bottema's Shoulders II
  6. On Bottema's Shoulders with a Ladder
  7. Friendly Kiepert's Perspectors
  8. Bottema Shatters Japan's Seclusion
  9. Rotations in Disguise
  10. Four Hinged Squares
  11. Four Hinged Squares, Solution with Complex Numbers
  12. Pythagoras' from Bottema's
  13. A Degenerate Case of Bottema's Configuration
  14. Properties of Flank Triangles
  15. Analytic Proof of Bottema's Theorem
  16. Yet Another Generalization of Bottema's Theorem
  17. Bottema with a Product of Rotations
  18. Bottema with Similar Triangles
  19. Bottema in Three Rotations
  20. Bottema's Point Sibling

|Activities| |Contact| |Front page| |Contents| |Geometry|

Copyright © 1996-2018 Alexander Bogomolny

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