The following transformation geometry theorem is sometimes very useful in problem solving.
Theorem
The sum (or composition) of two rotations a and b, respectively around centres A and B, is equal to a rotation a + b around another centre X. This centre X is located at a position where ∠XAB = a/2 and ∠ABX = b/2.
Illustration: The interactive diagram below dynamically illustrates the theorem.
Sum of Two Rotations Theorem
Challenge
Can you explain why (prove that) the above theorem is true?
Some Applications
1) An application of this theorem to an interesting, classic problem can be found at: Pirate Treasure Hunt.
This link also includes a related paper of mine which gives a proof the 'two rotations' theorem.
2) Other examples of the application of this theorem can be found in proofs of Van Aubel's theorem & some of its generalizations.
3) It can also be used in the proof of Pompe's Hexagon Theorem, or viewed as being equivalent to it.
Related Links
Pirate Treasure Hunt and a Generalization
Van Aubel's Theorem and some Generalizations
Pompe's Hexagon Theorem
Sum of Two Rotations Theorem
Fermat-Torricelli Point Generalization
Napoleon's Theorem: Generalizations & Converses
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Created by Michael de Villiers, 31 May 2022 with WebSketchpad; updated 20 Nov 2023.