“Equilateral Triangle”的概念、定义、习题解题方法及翻译-数学辞典

单词

Equilateral Triangle

释义

Equilateral Triangle

An equilateral triangle is a Triangle with all three sides of equal length . An equilateral triangle also hasthree equal 60° Angles.

An equilateral triangle can be constructed by Trisecting all three Angles ofany Triangle (Morley's Theorem). Napoleon's Theorem states that if three equilateral trianglesare drawn on the Legs of any Triangle (either all drawn inwards or outwards) and the centersof 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

(1)

(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).

The Altitude of an equilateral triangle is

(2)

where

is the side length, so the Area is

(3)

The Inradius , Circumradius , and Area can be computed directly from the formulas for ageneral regular Polygon with side length and sides,

			(4)
			(5)
			(6)

The Areas of the Incircle and Circumcircle are

			(7)
			(8)

Let any Rectangle be circumscribed about an Equilateral Triangle. Then

(9)

where

, and

are the Areas of the triangles in the figure (Honsberger 1985).

Begin with an arbitrary Triangle and find the Excentral Triangle. Then find the Excentral Triangleof that triangle, and so on. Then the resulting triangle approaches an equilateral triangle. The only RationalTriangle is the equilateral triangle (Conway and Guy 1996). A Polyhedron composed of only equilateral trianglesis known as a Deltahedron.

The smallest equilateral triangle which can be inscribed in a Unit Square (left figure) has side length and area

			(10)
			(11)

The largest equilateral triangle which can be inscribed (right figure) is oriented at an angle of 15° and has side length and area

			(12)
			(13)

(Madachy 1979).

See also Acute Triangle, Deltahedron, Equilic Quadrilateral, Fermat Point,Gyroelongated Square Dipyramid, Icosahedron, Isogonic Centers, Isosceles Triangle,Morley's Theorem, Octahedron, Pentagonal Dipyramid, Right Triangle, ScaleneTriangle, Snub Disphenoid, Tetrahedron, Triangle, Triangular Dipyramid,Triaugmented Triangular Prism, Viviani's Theorem

References

Beyer, W. H. (Ed.) CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 121, 1987.

Conway, J. H. and Guy, R. K. ``The Only Rational Triangle.'' In The Book of Numbers. New York: Springer-Verlag, pp. 201 and 228-239, 1996.

Dixon, R. Mathographics. New York: Dover, p. 33, 1991.

Gardner, M. Mathematical Carnival: A New Round-Up of Tantalizers and Puzzles from Scientific American. New York: Vintage Books, 1977.

Guy, R. K. ``Rational Distances from the Corners of a Square.'' §D19 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 181-185, 1994.

Honsberger, R. ``Equilateral Triangles.'' Ch. 3 in Mathematical Gems I. Washington, DC: Math. Assoc. Amer., 1973.

Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., pp. 19-21, 1985.

Madachy, J. S. Madachy's Mathematical Recreations. New York: Dover, pp. 115 and 129-131, 1979.

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