Triangle inequalities bound combinations of the sides, angles, and other geometric quantities of a triangle.
Let a triangle have angles 
, 
, and 
, then inequalities include
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(1)
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(2)
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(3)
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(4)
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(5)
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(6)
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(7)
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(8)
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(9)
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(10)
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(11)
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(12)
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(Siddons and Hughes 1929, p. 283), and
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(13)
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(Weisstein).
Another symmetric inequality is
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(14)
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with equality iff the triangle is an equilateral triangle (Trott 2006).
Trott (2006) uses quantifier elimination to establish bounds for weighted cosine sums of the angles of a triangle. For the first sum, his strict lower bound of 0 can be strengthened to 1, giving
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(15)
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where the upper bound is attained iff 
and 
. The stronger lower bound
follows from 
and 
.
For the second sum, Trott's bounds
of 0 and 4 are strict for nondegenerate triangles, so
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(16)
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For the first weighted sum, the lower bound is approached as 
tends to 
. For the second, the lower
bound and upper bound are approached as 
tends to 
and 
, respectively. None of these limiting bounds
is attained by a nondegenerate triangle.
For side lengths 
, 
, and 
, let 
be the semiperimeter,
the circumradius, 
the inradius, and 
the length of the triangle
median to the side of length 
. Two further inequalities are
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(17)
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with equality iff the triangle is an isosceles triangle with 
or a right triangle with
hypotenuse 
(Trott 2006). Also,
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(18)
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with equality iff the triangle is an equilateral triangle (Bjelica 1995,
Trott 2006). The second inequality follows from 
and 
. Another side-length
inequality is Padoa's
inequality.
