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An equilateral triangle is a triangle in ... of its base and height. The formula of the area of an ... of an equilateral triangle are 60°, the formula is as desired. ...
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]
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]
For acute triangles, the feet of the altitudes all fall on the triangle's sides (not extended). In an obtuse triangle (one with an obtuse angle ), the foot of the altitude to the obtuse-angled vertex falls in the interior of the opposite side, but the feet of the altitudes to the acute-angled vertices fall on the opposite extended side ...
The area of a triangle can be demonstrated, for example by means of the congruence of triangles, as half of the area of a parallelogram that has the same base length and height. A graphic derivation of the formula T = h 2 b {\displaystyle T={\frac {h}{2}}b} that avoids the usual procedure of doubling the area of the triangle and then halving it.
The inradius and circumradius formulas for an isosceles triangle may be derived from their formulas for arbitrary triangles. [31] The radius of the inscribed circle of an isosceles triangle with side length a {\displaystyle a} , base b {\displaystyle b} , and height h {\displaystyle h} is: [ 17 ]
There are many ways to prove Heron's formula, for example using trigonometry as below, or the incenter and one excircle of the triangle, [8] or as a special case of De Gua's theorem (for the particular case of acute triangles), [9] or as a special case of Brahmagupta's formula (for the case of a degenerate cyclic quadrilateral).
This formula can be derived by partitioning the n-sided polygon into n congruent isosceles triangles, and then noting that the apothem is the height of each triangle, and that the area of a triangle equals half the base times the height. The following formulations are all equivalent: