Napoleon problem

Napoleon (Buonaparte) is known as military stratege and imperator, but he was also a chess player and amateur mathematician. Several problems are called "Napoleon problem"

Construct using compass alone the center of a given circle

Given a circle without its center, construct the center using only a compass (and no straightedge).
Solution
There are other methods, among them the one from Napoleon himself (really from Mascheroni and carried back to France by Napoleon).

Construct with compass alone a square inscribed in a given circle

If the circle is given without the center, first construct the center A as above.
Choose any point = 1st vertex B on this circle.
Remains to construct E opposite to B on a diameter, then M and P with BM=BP=AB√2
Solution

Similar triangles on sides of given triangle ABC

Given any triangle ABC, draw the three triangles PBA, BMC and ACN directly similar to ABC (that is without flip), outside of ABC.
Prove that centroids R, S and T of these three triangles build a triangle similar to ABC.
(That is also true for any corresponding points in the three triangles, as the orthocenters etc...)

Equilateral triangles on a given ABC triangle

Draw the three equilateral triangles ABP, BCM and CAN on the sides of any given triangle ABC.
The three centers I, J and K of these triangles build an equilateral triangle (Napoleon theorem).
Animated JavaSketchpad (patience when loading and initializing Java engine).

Equilateral triangles on a segment - "Napoléon's hat"
It is a degenerate case of the previous problem, with C on AB.
Find the locus of the centroid of IJK when C moves along AB ? Hint

Proof of Napoleon's theorem :

 

 

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