# Napoleon's theorem

## Description

In geometry, Napoleon’s theorem states that if equilateral triangles are constructed on the sides of any triangle, either all outward, or all inward, the centres of those equilateral triangles themselves form an equilateral triangle.

The triangle thus formed is called the Napoleon triangle (inner and outer). The difference in area of these two triangles equals the area of the original triangle. Each of the three sides of the Napoleon triangle (inner) has a length that can be calculated by the lengths of the sides of the original triangle.

## Variables

N_{side} | Napoleon's triangle side (m) |

a | Side of the original triangle opposite to angle A (m) |

b | Side of the original triangle opposite to angle B (m) |

c | Side of the original triangle opposite to angle C (m) |