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Concept
Equilateral triangle
Summary
In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each other and are each 60°. It is also a regular polygon, so it is also referred to as a regular triangle.
Denoting the common length of the sides of the equilateral triangle as , we can determine using the Pythagorean theorem that:
The area is
The perimeter is
The radius of the circumscribed circle is
The radius of the inscribed circle is or
The geometric center of the triangle is the center of the circumscribed and inscribed circles
The altitude (height) from any side is
Denoting the radius of the circumscribed circle as R, we can determine using trigonometry that:
The area of the triangle is
Many of these quantities have simple relationships to the altitude ("h") of each vertex from the opposite side:
The area is
The height of the center from each side, or apothem, is
The radius of the circle circumscribing the three vertices is
The radius of the inscribed circle is
In an equilateral triangle, the altitudes, the angle bisectors, the perpendicular bisectors, and the medians to each side coincide.
A triangle that has the sides , , , semiperimeter , area , exradii , , (tangent to , , respectively), and where and are the radii of the circumcircle and incircle respectively, is equilateral if and only if any one of the statements in the following nine categories is true. Thus these are properties that are unique to equilateral triangles, and knowing that any one of them is true directly implies that we have an equilateral triangle.
(Blundon)
(Weitzenböck)
(Chapple-Euler)
Three kinds of cevians coincide, and are equal, for (and only for) equilateral triangles:
The three altitudes have equal lengths.
The three medians have equal lengths.
The three angle bisectors have equal lengths.
Every triangle center of an equilateral triangle coincides with its centroid, which implies that the equilateral triangle is the only triangle with no Euler line connecting some of the centers.
Official source
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