Triangle DFH is equilateral because all sides are equal and the angles are congruent due to the properties of squares inscribed in an equilateral triangle.
In the given configuration, since triangle ABC is equilateral, all its sides are equal in length. The squares ABDE, BCFG, and CAHI inscribed within the sides of the equilateral triangle share these equal side lengths. Considering square ABDE, DE is equal to the side length of triangle ABC.
Similarly, in square BCFG, CF is also equal to the side length of triangle ABC. Now, when we connect points D, F, and H, we form a straight line within square ABDE and square CAHI. Since DE is equal to HI, and DF and FH are sides of square ABDE, it follows that DF and FH are also equal. Therefore, triangle DFH has all sides equal, making it equilateral.