Үшбұрыштардың қабырғаларында салынған теңқабырғалы үшбұрыштар орталықтарын қосатын түзулер туралы

On lines connecting the centres of equilateral triangles built on the sides of a triangle

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 lines connecting the centres of those equilateral triangles themselves form an equilateral triangle. The triangle thus formed is called the inner or outer Napoleon triangle. The difference in the areas of the outer and inner Napoleon triangles equals the area of the original triangle. The theorem is often attributed to Napoleon Bonaparte (1769–1821). Some have suggested that it may date back to W. Rutherford's 1825 question published in The Ladies' Diary, four years after the French emperor's death, but the result is covered in three questions set in an examination for a Gold Medal at the University of Dublin in October, 1820, whereas Napoleon died the following May.

The centers of both the inner and outer Napoleon triangles coincide with the centroid of the original triangle. This coincidence was noted in Chambers's Encyclopaedia in 1867, as quoted above. The entry there is unsigned. P. G. Tait, then Professor of Natural Philosophy in the University of Edinburgh, is listed amongst the contributors, but J. U. Hillhouse, Mathematical Tutor also at the University of Edinburgh, appears amongst other literary gentlemen connected for longer or shorter times with the regular staff of the Encyclopaedia. However, in Section 189(e) of An Elementary Treatise on Quaternions, also in 1867, Tait treats the problem (in effect, echoing Davies' remarks in the Gentleman's Diary in 1831 with regard to Question 1265, but now in the setting of quaternions):
If perpendiculars be erected outwards at the middle points of the sides of a triangle, each being proportional to the corresponding side, the mean point of their extremities coincides with that of the original triangle. Find the ratio of each perpendicular to half the corresponding side of the old triangle that the new triangle may be equilateral. Tait concludes that the mean points of equilateral triangles erected outwardly on the sides of any triangle form an equilateral triangle. The discussion is retained in subsequent editions in 1873 and 1890, as well as in his further Introduction to Quaternions jointly with Philip Kelland in 1873.

If isosceles triangles with apex angles \tfrac{2k\pi}{n} are erected on the sides of an arbitrary n gon A0, and if this process is repeated with the n gon formed by the free apices of the triangles, but with a different value of k, and so on until all values 1 ≤ k ≤ n − 2 have been used (in arbitrary order), then a regular n gon An−2 is formed whose centroid coincides with the centroid of A0.

The centers of regular n gons constructed over 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.

Given a hexagon A1A2A3A4A5A6 with equilateral triangles constructed on the sides, either inwardly or outwardly, and the apexes of the equilateral triangles labelled Bi. If G1, G3, G5 are the respective centroids of △B6B1B2, △B2B3B4, △B4B5B6, then G1, G3, G5 form an equilateral triangle.

Given a hexagon ABCDEF with equilateral ∆'s ABG, DHC, IEF constructed on the alternate sides AB, CD and EF, either inwardly or outwardly. Let A1, B1, C1 be the centroids of ∆FGC, ∆BHE, and ∆DIA respectively, let A2, B2, C2 be the centroids of ∆DGE, ∆AHF, and ∆BIC respectively. Then ∆A1B1C1 and ∆A2B2C2 are equilateral triangles. (If, for example, we let points A and F coincide, as well as B and C, and D and E, then Dao Than Oai’s result reduces to Napoleon’s theorem).