The Nine Point Circle

Given three points A, B, C which are NOT collinear, a circle may be drawn passing through those points. The circle is centred at the point (O) where perpendicular bisectors of lines joining AB and BC intersect.

Napoleon's Theorem

Napoleon's theorem 

If equilateral triangles are constructed on the sides of any triangle, either all outward or all inward, the lines connecting the centroids 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 attributed to Napoleon Bonaparte (1769–1821) though this has been questioned.

This short video presents a visual proof - Youtube: Napoleon's theorem - proof

Further material:

Triangles (8) ... Proof by Ceva's Theorem of concurrence at the Fermat Point

Ceva's Theorem (and its converse) were covered in a previous post - Ceva's Theorem. 

In the earlier post on the Fermat Point, it was stated there is a concurrency at the Fermat Point but no proof was presented. As is often the case, there are various proofs. This post offers a proof using the converse of Ceva's Theorem.

Steps 1 and 2 establish two mathematical results which will be used in the proof.

Triangles (7) ... The Fermat Point

△ ABC is ANY acute triangle (i.e. all three angles are less than 90^o). Equilateral triangles are drawn on each side of  △ ABC. They are shown in the diagram as red △ BCY, green  △ ACZ, and blue  △ ABX.

Straight lines are then drawn joining A to Y, B to Z and C to X. Those lines intersect at F - the FERMAT point - named after Pierre de Fermat (1601 - 1665) who is probably better known for the so-called "Last Theorem" proved in the early 1990s by Professor Andrew Wiles.

The Fermat Point is sometimes called the Fermat - Torricelli point - (after Evangelista Torricelli 1608 - 1647).

The Fermat Point is the point in the triangle from which the sum of the distances to the vertices is a minimum. That is, where AF + BF + CF is smaller than the sum of the distances from any other point in the triangle to the vertices.

It can