In geometry, triangle inequalities are inequalities involving the parameters of triangles, that hold for every triangle, or for every triangle meeting certain conditions. The inequalities give an ordering of two different values: they are of the form “less than”, “less than or equal to”, “greater than”, or “greater than or equal to”. The parameters in a triangle inequality can be the side lengths, the semiperimeter, the angle measures, the values of trigonometric functions of those angles, the area of the triangle, the medians of the sides, the altitudes, the lengths of the internal angle bisectors from each angle to the opposite side, the perpendicular bisectors of the sides, the distance from an arbitrary point to another point, the inradius, the exradii, the circumradius, and/or other quantities.
Unless otherwise specified, this article deals with triangles in the Euclidean plane.
The parameters most commonly appearing in triangle inequalities are:
the side lengths a, b, and c;
the semiperimeter s = (a + b + c) / 2 (half the perimeter p);
the angle measures A, B, and C of the angles of the vertices opposite the respective sides a, b, and c (with the vertices denoted with the same symbols as their angle measures);
the values of trigonometric functions of the angles;
the area T of the triangle;
the medians ma, mb, and mc of the sides (each being the length of the line segment from the midpoint of the side to the opposite vertex);
the altitudes ha, hb, and hc (each being the length of a segment perpendicular to one side and reaching from that side (or possibly the extension of that side) to the opposite vertex);
the lengths of the internal angle bisectors ta, tb, and tc (each being a segment from a vertex to the opposite side and bisecting the vertex’s angle);
the perpendicular bisectors pa, pb, and pc of the sides (each being the length of a segment perpendicular to one side at its midpoint and reaching to one of the other sides);
the lengths of line segments with an endpoint at an arbitrary point P in the plane (for example, the length of the segment from P to vertex A is denoted PA or AP);
the inradius r (radius of the circle inscribed in the triangle, tangent to all three sides), the exradii ra, rb, and rc (each being the radius of an excircle tangent to side a, b, or c respectively and tangent to the extensions of the other two sides), and the circumradius R (radius of the circle circumscribed around the triangle and passing through all three vertices).
The basic triangle inequality is
{\displaystyle a
{\displaystyle \max(a,b,c)
{\displaystyle {\frac {3}{2}}\leq {\frac {a}{b+c}}+{\frac {b}{a+c}}+{\frac {c}{a+b}}<2,}{\displaystyle {\frac {3}{2}}\leq {\frac {a}{b+c}}+{\frac {b}{a+c}}+{\frac {c}{a+b}}<2,}
where the value of the right side is the lowest possible bound,[1]: p. 259 approached asymptotically as certain classes of triangles approach the degenerate case of zero area. The left inequality, which holds for all positive a, b, c, is Nesbitt’s inequality.
We have
{\displaystyle 3\left({\frac {a}{b}}+{\frac {b}{c}}+{\frac {c}{a}}\right)\geq 2\left({\frac {b}{a}}+{\frac {c}{b}}+{\frac {a}{c}}\right)+3.}3\left({\frac {a}{b}}+{\frac {b}{c}}+{\frac {c}{a}}\right)\geq 2\left({\frac {b}{a}}+{\frac {c}{b}}+{\frac {a}{c}}\right)+3.[2]: p.250, #82
{\displaystyle abc\geq (a+b-c)(a-b+c)(-a+b+c).\quad }abc\geq (a+b-c)(a-b+c)(-a+b+c).\quad [1]: p. 260
{\displaystyle {\frac {1}{3}}\leq {\frac {a{2}+b{2}+c{2}}{(a+b+c){2}}}<{\frac {1}{2}}.\quad }{\displaystyle {\frac {1}{3}}\leq {\frac {a{2}+b{2}+c{2}}{(a+b+c){2}}}<{\frac {1}{2}}.\quad }[1]: p. 261
{\displaystyle {\sqrt {a+b-c}}+{\sqrt {a-b+c}}+{\sqrt {-a+b+c}}\leq {\sqrt {a}}+{\sqrt {b}}+{\sqrt {c}}.}\sqrt{a+b-c} + \sqrt{a-b+c} + \sqrt{-a+b+c} \leq \sqrt{a}+\sqrt{b} + \sqrt{c}.[1]: p. 261
{\displaystyle a{2}b(a-b)+b{2}c(b-c)+c^{2}a(c-a)\geq 0.}a{2}b(a-b)+b{2}c(b-c)+c^{2}a(c-a)\geq 0.[1]: p. 261
If angle C is obtuse (greater than 90°) then
{\displaystyle a{2}+b{2}
if C is acute (less than 90°) then
{\displaystyle a{2}+b{2}>c{2}.}a{2}+b{2}>c{2}.
The in-between case of equality when C is a right angle is the Pythagorean theorem.
In general,[2]: p.1, #74
{\displaystyle a{2}+b{2}>{\frac {c{2}}{2}},}a{2}+b^{2}>{\frac {c^{2}}{2}},
with equality approached in the limit only as the apex angle of an isosceles triangle approaches 180°.
If the centroid of the triangle is inside the triangle’s incircle, then[3]: p. 153
{\displaystyle a^{2}<4bc,\quad b^{2}<4ac,\quad c{2}<4ab.}a{2}<4bc,\quad b^{2}<4ac,\quad c^{2}<4ab.
While all of the above inequalities are true because a, b, and c must follow the basic triangle inequality that the longest side is less than half the perimeter, the following relations hold for all positive a, b, and c:[1]: p.267
{\displaystyle {\frac {3abc}{ab+bc+ca}}\leq {\sqrt[{3}]{abc}}\leq {\frac {a+b+c}{3}},}{\frac {3abc}{ab+bc+ca}}\leq {\sqrt[ {3}]{abc}}\leq {\frac {a+b+c}{3}},
each holding with equality only when a = b = c. This says that in the non-equilateral case the harmonic mean of the sides is less than their geometric mean which in turn is less than their arithmetic mean.