# An Extra Triple of Equilateral Triangles for Napoleon

### What is this about?

### Equilateral Triangles in Napoleon's Configuration

Let $ABC',$ $BCA',$ and $CAB'$ be Napoleon's triangles constructed on the sides of $\Delta ABC.$ Define $A_B$ and $A_C$ as the intersections of $A'B$ and, respectively, $A'C$ with the circumcircle $C(ABC)$ of $\Delta ABC.$ Define similarly $B_A,B_C,C_A,C_B.$

The triangles $A_{B}B_{A}C,$ $AB_{C}C_{B},$ and $A_{C}BC_{A}$ are equilateral.

### Hint

The problem is very simple; it submits to chasing inscribed angles.

### Solution

Consider, for example, $\Delta AB_{C}C_{B}$ and, more specifically, its side $AB_{C}.$ This is a chord in $C(ABC).$

Since $\Delta CAB'$ is equilateral $\angle ACB_C=120^{\circ},$ meaning that side $AB_{C}$ subtends one third of the circumcircle. The same holds for the other sides and for the remaining two triangles.

### Acknowledgment

This problem has been posted by Dao Thanh Oai at the CutTheKnotMath facebook page.

### 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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