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triangle∗
Wkbj79†
2013-03-21 12:21:28
A triangle is a bounded planar region delimited by three straight lines, i.e.
it is a polygon with three angles.
In Euclidean geometry, 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 , a triangle with 2 equal sides is
called isosceles, and finally a triangle with no equal sides is called scalene. Notice
that an equilateral triangle is also isosceles, but there are isosceles triangles that
are not equilateral.
In Euclidean geometry, triangles can also be classified according to the size
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 angle, it is a right triangle. If the triangle has an obtuse
angle, it is an obtuse triangle.
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Area of a triangle
There are several ways to calculate a triangle’s area.
In hyperbolic and spherical geometry, the area of a triangle is equal to its
defect (measured in radians).
For the rest of this entry, only Euclidean geometry will be considered.
Many formulas for the area of a triangle exist. The most basic one is A =
1
bh, where b is its base and h is its height. Following is a derivation of another
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formula for the area of a triangle.
Let a, b, c be the sides and A, B, C the interior angles opposite to them. Let
ha , hb , hc be the heights drawn upon a, b, c respectively, r the inradius and R
a+b+c
the circumradius. Finally, let s =
be the semiperimeter. Then
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Area
=
=
=
=
=
bhb
chc
aha
=
=
2
2
2
ab sin C
bc sin A
ca sin B
=
=
2
2
2
abc
4R
sr
p
s(s − a)(s − b)(s − c)
The last formula 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
x1 y1 1
1
± x2 y2 1
2
x3 y3 1
(cf. the volume of tetrahedron).
Inequalities for the area are Weizenbock’s inequality and the HadwigerFinsler inequality.
Angles in a triangle
1. the sum of the angles in a triangle is π radians (180◦ )
2. sines law
3. cosines law
4. Mollweide’s equations
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Special geometric objects for a triangle
1. incenter
2. inscribed circle
3. circumcenter
4. circumscribed circle
5. centroid
6. orthocenter
7. Lemoine point, Lemoine circle
8. Gergonne point, Gergonne triangle
9. orthic triangle
10. pedal triangle
11. medial triangle
12. Euler Line
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