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 of 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
Airport Problem (Rethinking Proof activity))
Napoleon's Theorem (Rethinking Proof activity)
Mystery Transformation
Graph Enlargement/Reduction (Dilation): g(x) = af(x/a)
All parabola are similar
Parabola locus theorem
Interesting Locus Result
Vertical Line and Point Symmetries of Differentiable Functions
A Fundamental Theorem of Similarity
Transformations of Graphs
All cubic polynomial functions are affine equivalent
The point symmetry of a cubic polynomial (An illustration)
Point & Line Reflections in Concurrent Lines
Two Reflections Equal a Rotation
An Area Preserving Transformation: Shearing
A Surprising Constant Area Sum Involving Translating Figures
The affine invariance of the conics
External Links
Investigate Geometric Transformations as Functions (Geometric Functions)
Aimssec Lesson Activities (African Institute for Mathematical Sciences Schools Enrichment Centre)
UCT Mathematics Competition Training Material
SA Mathematics Olympiad Questions and worked solutions for past South African Mathematics Olympiad papers can be found at this link.
(Note, however, that prospective users will need to register and log in to be able to view past papers and solutions.)
Free Download of Geometer's Sketchpad & Associated Learning/Instructional Modules on Various Topics
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Created by Michael de Villiers, 31 May 2022 with WebSketchpad; updated 20 Nov 2023, 2 Oct 2026.