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,

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The areas of the incircle and circumcircle are

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Central triangles that are equilateral include the circumnormal triangle, circumtangential triangle, first Morley triangle, inner Napoleon triangle, outer Napoleon triangle, second Morley triangle, Stammler triangle, and third Morley triangle.

An equation giving an equilateral triangle with is given by

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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

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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

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

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(Madachy 1979).

Triangle line picking for points in an equilateral triangle with side lengths gives a mean line segment length of

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