# An Extra Triple of Equilateral Triangles for Napoleon

### 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.