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单词 Triangle
释义

triangle


A triangle is a planar region delimited by three lines, i.e. it is a polygonMathworldPlanetmathPlanetmath with three angles.

In Euclidean geometryMathworldPlanetmath, the angle sum of a triangle is always equal to 180. In the figure: A+B+C=180.

In hyperbolic geometry, the angle sum of a triangle is always strictly positive and strictly less than 180. In the figure: 0<A+B+C<180.

In spherical geometry, the angle sum of a triangle is always strictly greater than 180 and strictly less than 540. In the figure: 180<A+B+C<540.

Also in spherical geometry, a triangle has these additional requirements: It must be strictly contained in a hemisphere of the sphere that is serving as the model for spherical geometry, and all of its angles must have a measure strictly less that 180.

Triangles can be classified according to the number of their equal sides. So, a triangle with 3 equal sides is called equilateral (http://planetmath.org/RegularTriangle), a triangle with 2 equal sides is called isosceles, and finally a triangle with no equal sides is called scalene. Notice that an is also isosceles, but there are isosceles triangles that are not equilateral.

In Euclidean geometry, triangles can also be classified according to the of the greatest of its three (inner) angles. If the greatest of these is acute (and therefore all three are acute), the triangle is called an acute triangle. If the triangle has a right angleMathworldPlanetmathPlanetmath, it is a right triangle. If the triangle has an obtuse angle, it is an obtuse triangle.

Area of a triangle

There are several ways to a triangle’s area.

In hyperbolic and spherical , the area of a triangle is equal to its defect (measured in radians).

For the rest of this entry, only Euclidean geometry will beconsidered.

Many for the area of a triangle exist. The most basic one is A=12bh, where b is its base and h is its height. Following is a of another for the area of a triangle.

Let a,b,c be the sides and A,B,C the interior angles to them. Let ha,hb,hc be the heights drawn upon a,b,c respectively, r the inradiusMathworldPlanetmath and R the circumradiusMathworldPlanetmath. Finally, let s=a+b+c2 be the semiperimeter. Then

Area=aha2=bhb2=chc2
=absinC2=bcsinA2=casinB2
=abc4R
=sr
=s(s-a)(s-b)(s-c)

The last is known as Heron’s formula.

With the coordinates of the vertices  (x1,y1),  (x2,y2),  (x3,y3)  of the triangle, the area may be expressed as

±12|x1y11x2y21x3y31|

(cf. the volume of tetrahedronMathworldPlanetmathPlanetmath (http://planetmath.org/Tetrahedron)).

InequalitiesMathworldPlanetmath for the area are Weizenbock’s inequality and the Hadwiger-Finsler inequality.

Angles in a triangle

  1. 1.

    the sum of the angles in a triangle is π radians (180)

  2. 2.

    sines law

  3. 3.

    cosines law

  4. 4.

    Mollweide’s equations

Special geometric objects for a triangle

  1. 1.

    incenter

  2. 2.

    inscribed circle

  3. 3.

    circumcenter

  4. 4.

    circumscribed circle

  5. 5.

    centroidPlanetmathPlanetmath

  6. 6.

    orthocenterMathworldPlanetmath

  7. 7.

    Lemoine point, Lemoine circleMathworldPlanetmath

  8. 8.

    Gergonne pointMathworldPlanetmath, Gergonne triangle

  9. 9.

    orthic triangleMathworldPlanetmath

  10. 10.

    pedal triangleMathworldPlanetmath

  11. 11.

    medial triangleMathworldPlanetmath

  12. 12.

    Euler LineMathworldPlanetmath

Titletriangle
Canonical nameTriangle
Date of creation2013-03-22 11:43:51
Last modified on2013-03-22 11:43:51
OwnerWkbj79 (1863)
Last modified byWkbj79 (1863)
Numerical id54
AuthorWkbj79 (1863)
Entry typeDefinition
Classificationmsc 51-00
Classificationmsc 51M05
Classificationmsc 00A05
Classificationmsc 51M10
Classificationmsc 55-00
Classificationmsc 55-01
Related topicSinesLaw
Related topicEulerLine
Related topicMedian
Related topicPythagorasTheorem
Related topicHypotenuseMathworldPlanetmath
Related topicOrthocenter
Related topicOrthicTriangle
Related topicIsoscelesTriangle
Related topicCevasTheorem
Related topicCevian
Related topicSinesLawProof
Related topicFundamentalTheoremOnIsogonalLines
Related topicIncenter
Related topicEquilateralTriangle
Related topicTrigonometricVersionOfCevasTheorem
Related topicHeronsFo
Definesacute triangle
Definesright triangle
Definesobtuse triangle
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更新时间:2025/5/4 17:17:02