Short answers, pulled from the story.
Napoleon's theorem states that if equilateral triangles are constructed on the sides of any triangle, either all outward or all inward, the lines connecting the centers of those equilateral triangles form an equilateral triangle. The difference in area between the outer and inner Napoleon triangles equals the area of the original triangle.
Attribution is disputed. The theorem appeared as three consecutive problems in a University of Dublin gold medal examination in October 1820, and Napoleon died the following May. According to Howard Eves, the result was discovered by Napoleon's friend and adviser Lorenzo Mascheroni (1750-1800), who let the Emperor claim it. Napoleon's name was not connected to the theorem in print until Chambers's Encyclopaedia in 1867.
An early printed appearance is the Ladies' Diary of 1825, where William Rutherford of Woodburn posed the problem without mentioning Napoleon. The Dublin Problems volume published in 1823 recorded the theorem from the October 1820 gold medal examination. The result appeared with proof in James Thomson's Euclid textbook by 1834.
William Rutherford was a mathematician based at Woodburn who posed the theorem as a challenge question in the Ladies' Diary of 1825. He was a capable mathematician who could have proved the result himself; his reasons for asking others to demonstrate it are unknown. The Woodburn Problem Solving Group he led was notable enough to be described in a regional survey of Northumberland.
The Napoleon-Barlotti theorem generalizes Napoleon's result to n-gons. It states that the centers of regular n-gons constructed on the sides of an n-gon P form a regular n-gon if and only if P is an affine image of a regular n-gon.
The centers of both the inner and outer Napoleon triangles coincide with the centroid of the original triangle. This property was noted in Chambers's Encyclopaedia in 1867 and treated by P. G. Tait, Professor of Natural Philosophy at the University of Edinburgh, in his Elementary Treatise on Quaternions in the same year.