# Napoleon on Hinges

*Napoleon's theorem* claims that the centers A', B', C', of the equilateral triangles A''BC, AB''C, ABC'', erected on the sides (either all inwardly or all outwardly) of a given triangle ABC form an equilateral triangle. The applet below serves to illustrate a very simple proof of this result.

Use the scroll bar at the bottom of the applet to rotate some triangles 120°. Which triangles are get rotated depends on which of the three boxes Swivel at "B, C", "A, C", or "A, B", is checked. Rotations are actually around A', B', or C'.

What if applet does not run? |

The proof below is a slight modification of a recently published proof by Alex Anderson (the proof has appeared a decade earlier in a Russian popular math and physics journal Kvant.) Below we assume that none of the angles in ΔABC exceeds 120°. The proof will have to be modified otherwise.

Assume the box "B, C" is checked so that we rotate triangles A'BC' (around C') and A'B'C (around B'). B and C are rotated onto A (because triangles ABC' and AB'C are isosceles with the apex angles at C' and B', respectively, equal to 120°.) Now, let's check the angles:

∠A'BC' | = ∠ABC + ∠ABC' + ∠CBA' | |

= ∠ABC + 30° + 30° | ||

= ∠ABC + 60°. |

Similarly, ∠A'CB' = ∠ACB + 60° and ∠B'AC' = ∠BAC + 60°.

At the end of rotations when triangles A'BC' and A'B'C meet at A, the sum of angles at A is exactly 360°:

∠A'BC' + ∠ A'CB' + ∠ B'AC' | = ∠ABC + ∠ACB + ∠BAC + 3×60° | |

= 360°. |

This means that, after the rotation, the sides A'B and A'C coalesce into a segment which we denote AA*. Observe that by SSS,

As we just saw, Napoleon's triangle A'B'C' is the union of the triangles congruent to triangles AB'C', A'BC', and A'B'C. This dissection can be obtained by reflecting A in B'C', B in A'C', and C in A'B', telling us that all three coincide. The common point, say X, is the Fermat point of ΔABC. Indeed,

XB' = AB' = CB' = B'B'',

Implying that X is on the circumcircle of ΔAB''C. Similarly, it is on the circumcircles of triangles ABC'' and A''BC and hence is their common point.

### References

- A. Anderson,
__A New Proof for Napoleon's Theorem__,*Mathematical Reflections*3 (2007) - A. Savin,
__Geometric Surprises__,*Kvant*2 (1996) (in Russian)

### Napoleon's Theorem

- Napoleon's Theorem
- A proof with complex numbers
- A second proof with complex numbers
- A third proof with complex numbers
- Napoleon's Theorem, Two Simple Proofs
- Napoleon's Theorem via Inscribed Angles
- A Generalization
- Douglas' Generalization
- Napoleon's Propeller
- Napoleon's Theorem by Plane Tessellation
- Fermat's point
- Kiepert's theorem
- Lean Napoleon's Triangles
- Napoleon's Theorem by Transformation
- Napoleon's Theorem via Two Rotations
- Napoleon on Hinges
- Napoleon on Hinges in GeoGebra
- Napoleon's Relatives
- Napoleon-Barlotti Theorem
- Some Properties of Napoleon's Configuration
- Fermat Points and Concurrent Euler Lines I
- Fermat Points and Concurrent Euler Lines II
- Escher's Theorem
- Circle Chains on Napoleon Triangles
- Napoleon's Theorem by Vectors and Trigonometry
- An Extra Triple of Equilateral Triangles for Napoleon
- Joined Common Chords of Napoleon's Circumcircles
- Napoleon's Hexagon
- Fermat's Hexagon
- Lighthouse at Fermat Points
- Midpoint Reciprocity in Napoleon's Configuration
- Another Equilateral Triangle in Napoleon's Configuration
- Yet Another Analytic Proof of Napoleon's Theorem
- Leo Giugiuc's Proof of Napoleon's Theorem
- Gregoie Nicollier's Proof of Napoleon's Theorem
- Fermat Point Several Times Over

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