Timeless Theorems of Mathematics/Napoleon's theorem

thumb| is an equilateral triangle The Napoleon's theorem states that if equilateral triangles are constructed on the sides of a triangle, either all outward or all inward, the lines connecting the centers of those equilateral triangles themselves form an equilateral triangle. That means, for a triangle , if three equilateral triangles are constructed on the sides of the triangle, such as , and either all outward or all inward, the three lines connecting the centers of the three triangles, , and construct an equilateral triangle . Proof thumb|A trigonomatric proof of the Napoleon's theorem. Let, a triangle. Here, three equilateral triangles are constructed, , and and the centroids of the triangles are , and respectively. Here, , , , , and . Therefore, the area of the triangle , For our proof, we will be working with one equilateral triangle, as three of the triangles are similar (equilateral). A median of is , where and . and, as is a equilateral triangle, . Here, . As the centroid of a triangle divides a median of the triangle as ratio, then . Similarly, . According to the Law of Cosines, (for ) and for , [According to the law of cosines for ] Therefore, In the same way, we can prove, and . Thus, . is an equilateral triangle. [Proved] 100%