Equilateral triangle Equilateral triangle In geometry, an equilateral triangle is a triangle in which all three sides are equal. 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 "a", we can determine using the Pythagorean theorem that: Denoting the radius of the circumscribed circle as "R", we can determine using trigonometry that: Many of these quantities
Equilateral triangle the location of "P". Pompeiu's theorem states that, if "P" is an arbitrary point in an equilateral triangle "ABC", then there exists a triangle with sides of lengths "PA", "PB", and "PC". That is, "PA", "PB", and "PC" satisfy the triangle inequality that any two of them sum to at least as great as the third. By Euler's inequality, the equilateral triangle has the smallest ratio "R"/"r" of the circumradius to the inradius of any triangle: specifically, "R"/"r" = 2. The triangle of largest area of all those inscribed in a given circle is equilateral; and the triangle of smallest