Equilateral triangle

Many of these quantities have simple relationships to the altitude ("h") of each vertex from the opposite side:

In an equilateral triangle, the altitudes, the angle bisectors, the perpendicular bisectors, and the medians to each side coincide.

For any triangle, the three medians partition the triangle into six smaller triangles.

Napoleon's theorem states that, if equilateral triangles are constructed on the sides of any triangle, either all outward, or all inward, the centers of those equilateral triangles themselves form an equilateral triangle.

There are numerous triangle inequalities that hold with equality if and only if the triangle is equilateral.

Equilateral triangles are the only triangles whose Steiner inellipse is a circle (specifically, it is the incircle).

An alternative method is to draw a circle with radius r, place the point of the compass on the circle and draw another circle with the same radius. The two circles will intersect in two points. An equilateral triangle can be constructed by taking the two centers of the circles and either of the points of intersection.

Equilateral triangles have frequently appeared in man made constructions: