Short answers, pulled from the story.
Napoleon's theorem states that connecting the centers of equilateral triangles built on the sides of any triangle always results in an equilateral triangle. This geometric fact holds true regardless of the original triangle's shape or size whether the small triangles point inward or outward.
The theorem bears the name of Napoleon Bonaparte who lived from 1769 to 1821 but Howard Eves suggests the discovery actually belongs to Lorenzo Mascheroni. A question appeared in W. Rutherford's 1825 publication called The Ladies' Diary four years after Napoleon died while examination papers from the University of Dublin in October 1820 contain similar problems before the French emperor passed away in May 1821.
A clockwise rotation of 30 degrees combined with a homothety ratio transforms one segment into another while a counterclockwise rotation of 30 degrees achieves the same transformation on the opposite side. These spiral similarities imply the angle between them must be exactly 60 degrees which proves the equilateral nature of the resulting figure.
The difference in area between the outer and inner versions equals the area of the starting triangle. Weitzenböck's inequality states equality occurs only when the starting triangle is already equilateral and algebraically the inner triangle is retrograde so its signed area becomes negative of that expression.
Centers of both inner and outer Napoleon triangles coincide with the centroid of the original figure as noted by P.G. Tait in Chambers's Encyclopaedia published in 1867. Perpendiculars erected at midpoints of sides prove the coincidence of these central points and mean points of outwardly erected equilateral triangles form an equilateral triangle.