# Equilateral Triangle

An equilateral triangle is a triangle with all three sides of equal length , corresponding to what could also be known as a “regular” triangle. An equilateral triangle is therefore a special case of an isosceles triangle having not just two, but all three sides equal. An equilateral triangle also has three equal  angles.

The altitude  of an equilateral triangle is

 (1)

where  is the side length, so the area is

 (2)

The inradius circumradius , and area  can be computed directly from the formulas for a general regular polygon with side length  and  sides,

 (3) (4) (5) (6) (7) (8) (9) (10)

The areas of the incircle and circumcircle are

 (11) (12) (13) (14)

Central triangles that are equilateral include the circumnormal trianglecircumtangential trianglefirst Morley triangleinner Napoleon triangleouter Napoleon trianglesecond Morley triangleStammler triangle, and third Morley triangle.

An equation giving an equilateral triangle with  is given by

 (15)

Geometric construction of an equilateral consists of drawing a diameter of a circle  and then constructing its perpendicular bisector . Bisect  in point , and extend the line  through . The resulting figure  is then an equilateral triangle. An equilateral triangle may also be constructed from the intersections of the angle trisectors of the three interior angles of any triangles (Morley’s theorem).

Napoleon’s theorem states that if three equilateral triangles are drawn on the legs of any triangle (either all drawn inwards or outwards) and the centers of these triangles are connected, the result is another equilateral triangle.

Given the distances of a point from the three corners of an equilateral triangle, , and , the length of a side  is given by

 (16)

(Gardner 1977, pp. 56-57 and 63). There are infinitely many solutions for which , and  are integers. In these cases, one of , and  is divisible by 3, one by 5, one by 7, and one by 8 (Guy 1994, p. 183).

Begin with an arbitrary triangle and find the excentral triangle. Then find the excentral triangle of that triangle, and so on. Then the resulting triangle approaches an equilateral triangle. The only rational triangle is the equilateral triangle (Conway and Guy 1996). A polyhedron composed of only equilateral triangles is known as a deltahedron.

Let any rectangle be circumscribed about an equilateral triangle. Then

 (17)

where , and  are the areas of the triangles in the figure (Honsberger 1985).

The smallest equilateral triangle which can be inscribed in a unit square (left figure) has side length and area

 (18) (19)

The largest equilateral triangle which can be inscribed (right figure) is oriented at an angle of  and has side length and area

 (20) (21)

(Madachy 1979).

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