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.
Main parameters and notation 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).
Side lengths The basic triangle inequality is
a ≤ b + c , b ≤ c + a , c ≤ a + b {\displaystyle a\leq b+c,\quad b\leq c+a,\quad c\leq a+b}
or equivalently
max ( a , b , c ) ≤ s . {\displaystyle \max(a,b,c)\leq s.}
In addition,
3 2 ≤ a b + c + b a + c + 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, 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
3 ( a b + b c + c a ) ≥ 2 ( b a + c b + a c ) + 3. {\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.}
a b c ≥ ( a + b − c ) ( a − b + c ) ( − a + b + c ) . {\displaystyle abc\geq (a+b-c)(a-b+c)(-a+b+c).\quad }
1 3 ≤ a 2 + b 2 + c 2 ( a + b + c ) 2 < 1 2 . {\displaystyle {\frac {1}{3}}\leq {\frac {a^{2}+b^{2}+c^{2}}{(a+b+c)^{2}}}<{\frac {1}{2}}.\quad }
a + b − c + a − b + c + − a + b + c ≤ a + b + c . {\displaystyle {\sqrt {a+b-c}}+{\sqrt {a-b+c}}+{\sqrt {-a+b+c}}\leq {\sqrt {a}}+{\sqrt {b}}+{\sqrt {c}}.}
a 2 b ( a − b ) + b 2 c ( b − c ) + c 2 a ( c − a ) ≥ 0. {\displaystyle a^{2}b(a-b)+b^{2}c(b-c)+c^{2}a(c-a)\geq 0.}
If angle C is obtuse (greater than 90°) then
a 2 + b 2 < c 2 ; {\displaystyle a^{2}+b^{2}<c^{2};}
if C is acute (less than 90°) then
a 2 + b 2 > c 2 . {\displaystyle a^{2}+b^{2}>c^{2}.}
The in-between case of equality when C is a right angle is the Pythagorean theorem. In general,
a 2 + b 2 > c 2 2 , {\displaystyle 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
a 2 < 4 b c , b 2 < 4 a c , c 2 < 4 a b . {\displaystyle a^{2}<4bc,\quad b^{2}<4ac,\quad c^{2}<4ab.}
Equivalently, if a , b , c {\displaystyle a,b,c} constitute the sides of a triangle and the triangle's centroid is inside the incircle then the equation a x 2 + b x + c = 0 {\displaystyle ax^{2}+bx+c=0} has no real roots. 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:
3 a b c a b + b c + c a ≤ a b c 3 ≤ a + b + c 3 ≤ a 2 + b 2 + c 2 3 , {\displaystyle {\frac {3abc}{ab+bc+ca}}\leq {\sqrt[{3}]{abc}}\leq {\frac {a+b+c}{3}}\leq {\sqrt {\frac {a^{2}+b^{2}+c^{2}}{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, and which in turn is less than their quadratic mean.
Angles
cos A + cos B + cos C ≤ 3 2 . {\displaystyle \cos A+\cos B+\cos C\leq {\frac {3}{2}}.}
( 1 − cos A ) ( 1 − cos B ) ( 1 − cos C ) ≥ cos A ⋅ cos B ⋅ cos C . {\displaystyle (1-\cos A)(1-\cos B)(1-\cos C)\geq \cos A\cdot \cos B\cdot \cos C.}
cos 4 A 2 + cos 4 B 2 + cos 4 C 2 ≤ s 3 2 a b c {\displaystyle \cos ^{4}{\frac {A}{2}}+\cos ^{4}{\frac {B}{2}}+\cos ^{4}{\frac {C}{2}}\leq {\frac {s^{3}}{2abc}}}
for semi-perimeter s, with equality only in the equilateral case.
a + b + c ≥ 2 b c cos A + 2 c a cos B + 2 a b cos C . {\displaystyle a+b+c\geq 2{\sqrt {bc}}\cos A+2{\sqrt {ca}}\cos B+2{\sqrt {ab}}\cos C.}
sin A + sin B + sin C ≤ 3 3 2 . {\displaystyle \sin A+\sin B+\sin C\leq {\frac {3{\sqrt {3}}}{2}}.}
sin 2 A + sin 2 B + sin 2 C ≤ 9 4 . {\displaystyle \sin ^{2}A+\sin ^{2}B+\sin ^{2}C\leq {\frac {9}{4}}.}
sin A ⋅ sin B ⋅ sin C ≤ ( sin A + sin B + sin C 3 ) 3 ≤ ( sin A + B + C 3 ) 3 = sin 3 ( π 3 ) = 3 3 8 . {\displaystyle \sin A\cdot \sin B\cdot \sin C\leq \left({\frac {\sin A+\sin B+\sin C}{3}}\right)^{3}\leq \left(\sin {\frac {A+B+C}{3}}\right)^{3}=\sin ^{3}\left({\frac {\pi }{3}}\right)={\frac {3{\sqrt {3}}}{8}}.}
sin A + sin B ⋅ sin C ≤ φ {\displaystyle \sin A+\sin B\cdot \sin C\leq \varphi }
where φ = 1 + 5 2 , {\displaystyle \varphi ={\frac {1+{\sqrt {5}}}{2}},} the golden ratio.
sin A 2 ⋅ sin B 2 ⋅ sin C 2 ≤ 1 8 . {\displaystyle \sin {\frac {A}{2}}\cdot \sin {\frac {B}{2}}\cdot \sin {\frac {C}{2}}\leq {\frac {1}{8}}.}
tan 2 A 2 + tan 2 B 2 + tan 2 C 2 ≥ 1. {\displaystyle \tan ^{2}{\frac {A}{2}}+\tan ^{2}{\frac {B}{2}}+\tan ^{2}{\frac {C}{2}}\geq 1.}
cot A + cot B + cot C ≥ 3 . {\displaystyle \cot A+\cot B+\cot C\geq {\sqrt {3}}.}
sin A ⋅ cos B + sin B ⋅ cos C + sin C ⋅ cos A ≤ 3 3 4 . {\displaystyle \sin A\cdot \cos B+\sin B\cdot \cos C+\sin C\cdot \cos A\leq {\frac {3{\sqrt {3}}}{4}}.}
For circumradius R and inradius r we have
max ( sin A 2 , sin B 2 , sin C 2 ) ≤ 1 2 ( 1 + 1 − 2 r R ) , {\displaystyle \max \left(\sin {\frac {A}{2}},\sin {\frac {B}{2}},\sin {\frac {C}{2}}\right)\leq {\frac {1}{2}}\left(1+{\sqrt {1-{\frac {2r}{R}}}}\right),}
with equality if and only if the triangle is isosceles with apex angle greater than or equal to 60°; and
min ( sin A 2 , sin B 2 , sin C 2 ) ≥ 1 2 ( 1 − 1 − 2 r R ) , {\displaystyle \min \left(\sin {\frac {A}{2}},\sin {\frac {B}{2}},\sin {\frac {C}{2}}\right)\geq {\frac {1}{2}}\left(1-{\sqrt {1-{\frac {2r}{R}}}}\right),}
with equality if and only if the triangle is isosceles with apex angle less than or equal to 60°. We also have
r R − 1 − 2 r R ≤ cos A ≤ r R + 1 − 2 r R {\displaystyle {\frac {r}{R}}-{\sqrt {1-{\frac {2r}{R}}}}\leq \cos A\leq {\frac {r}{R}}+{\sqrt {1-{\frac {2r}{R}}}}}
and likewise for angles B, C, with equality in the first part if the triangle is isosceles and the apex angle is at least 60° and equality in the second part if and only if the triangle is isosceles with apex angle no greater than 60°. Further, any two angle measures A and B opposite sides a and b respectively are related according to
A > B if and only if a > b , {\displaystyle A>B\quad {\text{if and only if}}\quad a>b,}
which is related to the isosceles triangle theorem and its converse, which state that A = B if and only if a = b. By Euclid's exterior angle theorem, any exterior angle of a triangle is greater than either of the interior angles at the opposite vertices:
180 ∘ − A > max ( B , C ) . {\displaystyle 180^{\circ }-A>\max(B,C).}
If a point D is in the interior of triangle ABC, then
∠ B D C > ∠ A . {\displaystyle \angle BDC>\angle A.}
For an acute triangle we have
cos 2 A + cos 2 B + cos 2 C < 1 , {\displaystyle \cos ^{2}A+\cos ^{2}B+\cos ^{2}C<1,}
with the reverse inequality holding for an obtuse triangle. Furthermore, for non-obtuse triangles we have
2 R + r R ≤ 2 ( cos ( A − C 2 ) + cos ( B 2 ) ) {\displaystyle {\frac {2R+r}{R}}\leq {\sqrt {2}}\left(\cos \left({\frac {A-C}{2}}\right)+\cos \left({\frac {B}{2}}\right)\right)}
with equality if and only if it is a right triangle with hypotenuse AC.
Area Weitzenböck's inequality is, in terms of area T,
a 2 + b 2 + c 2 ≥ 4 3 ⋅ T , {\displaystyle a^{2}+b^{2}+c^{2}\geq 4{\sqrt {3}}\cdot T,}
with equality only in the equilateral case. This is a corollary of the Hadwiger–Finsler inequality, which is
a 2 + b 2 + c 2 ≥ ( a − b ) 2 + ( b − c ) 2 + ( c − a ) 2 + 4 3 ⋅ T . {\displaystyle a^{2}+b^{2}+c^{2}\geq (a-b)^{2}+(b-c)^{2}+(c-a)^{2}+4{\sqrt {3}}\cdot T.}
Also,
a b + b c + c a ≥ 4 3 ⋅ T {\displaystyle ab+bc+ca\geq 4{\sqrt {3}}\cdot T}
and
T ≤ a b c 2 a + b + c a 3 + b 3 + c 3 + a b c ≤ 1 4 3 ( a + b + c ) 3 ( a b c ) 4 a 3 + b 3 + c 3 6 ≤ 3 4 ( a b c ) 2 3 . {\displaystyle T\leq {\frac {abc}{2}}{\sqrt {\frac {a+b+c}{a^{3}+b^{3}+c^{3}+abc}}}\leq {\frac {1}{4}}{\sqrt[{6}]{\frac {3(a+b+c)^{3}(abc)^{4}}{a^{3}+b^{3}+c^{3}}}}\leq {\frac {\sqrt {3}}{4}}(abc)^{\frac {2}{3}}.}
From the rightmost upper bound on T, using the arithmetic-geometric mean inequality, is obtained the isoperimetric inequality for triangles:
T ≤ 3 36 ( a + b + c ) 2 = 3 9 s 2 {\displaystyle T\leq {\frac {\sqrt {3}}{36}}(a+b+c)^{2}={\frac {\sqrt {3}}{9}}s^{2}} for semiperimeter s. This is sometimes stated in terms of perimeter p as
p 2 ≥ 12 3 ⋅ T , {\displaystyle p^{2}\geq 12{\sqrt {3}}\cdot T,}
with equality for the equilateral triangle. This is strengthened by
T ≤ 3 4 ( a b c ) 2 3 . {\displaystyle T\leq {\frac {\sqrt {3}}{4}}\left(abc\right)^{\frac {2}{3}}.}
Bonnesen's inequality also strengthens the isoperimetric inequality:
π 2 ( R − r ) 2 ≤ ( a + b + c ) 2 − 4 π T . {\displaystyle \pi ^{2}(R-r)^{2}\leq (a+b+c)^{2}-4\pi T.}
We also have
9 a b c a + b + c ≥ 4 3 ⋅ T {\displaystyle {\frac {9abc}{a+b+c}}\geq 4{\sqrt {3}}\cdot T} with equality only in the equilateral case;
38 T 2 ≤ 2 s 4 − a 4 − b 4 − c 4 {\displaystyle 38T^{2}\leq 2s^{4}-a^{4}-b^{4}-c^{4}}
for semiperimeter s; and
1 a + 1 b + 1 c < s T . {\displaystyle {\frac {1}{a}}+{\frac {1}{b}}+{\frac {1}{c}}<{\frac {s}{T}}.}
Ono's inequality for acute triangles (those with all angles less than 90°) is
27 ( b 2 + c 2 − a 2 ) 2 ( c 2 + a 2 − b 2 ) 2 ( a 2 + b 2 − c 2 ) 2 ≤ ( 4 T ) 6 . {\displaystyle 27(b^{2}+c^{2}-a^{2})^{2}(c^{2}+a^{2}-b^{2})^{2}(a^{2}+b^{2}-c^{2})^{2}\leq (4T)^{6}.}
The area of the triangle can be compared to the area of the incircle:
Area of incircle Area of triangle ≤ π 3 3 {\displaystyle {\frac {\text{Area of incircle}}{\text{Area of triangle}}}\leq {\frac {\pi }{3{\sqrt {3}}}}}
with equality only for the equilateral triangle. If an inner triangle is inscribed in a reference triangle so that the inner triangle's vertices partition the perimeter of the reference triangle into equal length segments, the ratio of their areas is bounded by
Area of inscribed triangle Area of reference triangle ≤ 1 4 . {\displaystyle {\frac {\text{Area of inscribed triangle}}{\text{Area of reference triangle}}}\leq {\frac {1}{4}}.}
Let the interior angle bisectors of A, B, and C meet the opposite sides at D, E, and F. Then
3 a b c 4 ( a 3 + b 3 + c 3 ) ≤ Area of triangle D E F Area of triangle A B C ≤ 1 4 . {\displaystyle {\frac {3abc}{4(a^{3}+b^{3}+c^{3})}}\leq {\frac {{\text{Area of triangle}}\,DEF}{{\text{Area of triangle}}\,ABC}}\leq {\frac {1}{4}}.}
A line through a triangle’s median splits the area such that the ratio of the smaller sub-area to the original triangle’s area is at least 4/9.
Medians and centroid The three medians m a , m b , m c {\displaystyle m_{a},\,m_{b},\,m_{c}} of a triangle each connect a vertex with the midpoint of the opposite side, and the sum of their lengths satisfies
3 4 ( a + b + c ) < m a + m b + m c < a + b + c . {\displaystyle {\frac {3}{4}}(a+b+c)<m_{a}+m_{b}+m_{c}<a+b+c.}
Moreover,
( m a a ) 2 + ( m b b ) 2 + ( m c c ) 2 ≥ 9 4 , {\displaystyle \left({\frac {m_{a}}{a}}\right)^{2}+\left({\frac {m_{b}}{b}}\right)^{2}+\left({\frac {m_{c}}{c}}\right)^{2}\geq {\frac {9}{4}},}
with equality only in the equilateral case, and for inradius r,
m a m b m c m a 2 + m b 2 + m c 2 ≥ r . {\displaystyle {\frac {m_{a}m_{b}m_{c}}{m_{a}^{2}+m_{b}^{2}+m_{c}^{2}}}\geq r.}
If we further denote the lengths of the medians extended to their intersections with the circumcircle as Ma , Mb , and Mc , then
M a m a + M b m b