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For example, since the perimeter of an isosceles triangle is the sum of its two legs and base, the equilateral triangle is formulated as three times its side. [ 3 ] [ 4 ] The internal angle of an equilateral triangle are equal, 60°. [ 5 ]
The Koch snowflake (also known as the Koch curve, Koch star, or Koch island [1] [2]) is a fractal curve and one of the earliest fractals to have been described. It is based on the Koch curve, which appeared in a 1904 paper titled "On a Continuous Curve Without Tangents, Constructible from Elementary Geometry" [3] by the Swedish mathematician Helge von Koch.
with equality only for the equilateral triangle. [11] If an inner triangle is inscribed in a reference triangle so that the inner triangle's vertices partition the perimeter of the reference triangle into equal length segments, the ratio of their areas is bounded by [9]: p. 138
Triangles have many types based on the length of the sides and the angles. A triangle whose sides are all the same length is an equilateral triangle, [3] a triangle with two sides having the same length is an isosceles triangle, [4] [a] and a triangle with three different-length sides is a scalene triangle. [7]
There are many ways to prove Heron's formula, for example using trigonometry as below, or the incenter and one excircle of the triangle, [7] or as a special case of De Gua's theorem (for the particular case of acute triangles), [8] or as a special case of Brahmagupta's formula (for the case of a degenerate cyclic quadrilateral).
The three line segments ¯, ¯ and ¯ are called the splitters of the triangle; they each bisect the perimeter of the triangle, [citation needed] ¯ + ¯ = ¯ + ¯ = (¯ + ¯ + ¯). The splitters intersect in a single point, the triangle's Nagel point N a {\displaystyle N_{a}} (or triangle center X 8 ).
For any interior point P, the sum of the lengths of the perpendiculars s + t + u equals the height of the equilateral triangle.. Viviani's theorem, named after Vincenzo Viviani, states that the sum of the shortest distances from any interior point to the sides of an equilateral triangle equals the length of the triangle's altitude. [1]
The center of the circle lies on the symmetry axis of the triangle, this distance above the base. An isosceles triangle has the largest possible inscribed circle among the triangles with the same base and apex angle, as well as also having the largest area and perimeter among the same class of triangles. [31]