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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 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 mediansm_{a}, m_{b}, and m_{c} of the sides (each being the length of the line segment from the midpoint of the side to the opposite vertex);
the altitudesh_{a}, h_{b}, and h_{c} (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 bisectorst_{a}, t_{b}, and t_{c} (each being a segment from a vertex to the opposite side and bisecting the vertex's angle);
the perpendicular bisectorsp_{a}, p_{b}, and p_{c} 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 inradiusr (radius of the circleinscribed in the triangle, tangent to all three sides), the exradiir_{a}, r_{b}, and r_{c} (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 circumradiusR (radius of the circle circumscribed around the triangle and passing through all three vertices).
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.
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}
$a^{2}<4bc,\quad b^{2}<4ac,\quad c^{2}<4ab.$
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}
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 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°.^{[7]}^{:Prop. 5}
Further, any two angle measures A and B opposite sides a and b respectively are related according to^{[1]}^{:p. 264}
$A>B\quad {\text{if and only if}}\quad a>b,$
which is related to the isosceles triangle theorem and its converse, which state that A = B if and only if a = b.
The area of the triangle can be compared to the area of the incircle:
${\frac {\text{Area of incircle}}{\text{Area of triangle}}}\leq {\frac {\pi }{3{\sqrt {3}}}}$
with equality only for the equilateral triangle.^{[11]}
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^{[9]}^{:p. 138}
${\frac {\text{Area of inscribed triangle}}{\text{Area of reference triangle}}}\leq {\frac {1}{4}}.$
Let the interior angle bisectors of A, B, and C meet the opposite sides at D, E, and F. Then^{[2]}^{:p.18,#762}
${\frac {3abc}{4(a^{3}+b^{3}+c^{3})}}\leq {\frac {{\text{Area of triangle}}\,DEF}{{\text{Area of triangle}}\,ABC}}\leq {\frac {1}{4}}.$
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.^{[12]}
Medians and centroid
The three medians$m_{a},\,m_{b},\,m_{c}$ of a triangle each connect a vertex with the midpoint of the opposite side, and the sum of their lengths satisfies^{[1]}^{:p. 271}
If we further denote the lengths of the medians extended to their intersections with the circumcircle as M_{a} ,
M_{b} , and M_{c} , then^{[2]}^{:p.16,#689}
The internal angle bisectors are segments in the interior of the triangle reaching from one vertex to the opposite side and bisecting the vertex angle into two equal angles. The angle bisectors t_{a} etc. satisfy
$t_{a}+t_{b}+t_{c}\leq {\frac {3}{2}}(a+b+c)$
in terms of the sides, and
$h_{a}\leq t_{a}\leq m_{a}$
in terms of the altitudes and medians, and likewise for t_{b} and t_{c} .^{[1]}^{:pp. 271-3} Further,^{[2]}^{:p.224,#132}
These inequalities deal with the lengths p_{a} etc. of the triangle-interior portions of the perpendicular bisectors of sides of the triangle. Denoting the sides so that $a\geq b\geq c,$ we have^{[20]}
$p_{a}\geq p_{b}$
and
$p_{c}\geq p_{b}.$
Segments from an arbitrary point
Interior point
Consider any point P in the interior of the triangle, with the triangle's vertices denoted A, B, and C and with the lengths of line segments denoted PA etc. We have^{[1]}^{:pp. 275-7}
$2(PA+PB+PC)>AB+BC+CA>PA+PB+PC,$
and more strongly than the second of these inequalities is^{[1]}^{:p. 278}
with equality in the equilateral case. More strongly, Barrow's inequality states that if the interior bisectors of the angles at interior point P (namely, of ?APB, ?BPC, and ?CPA) intersect the triangle's sides at U, V, and W, then^{[23]}
${\frac {PA+PB+PC}{PU+PV+PW}}\geq 2.$
Also stronger than the Erd?s-Mordell inequality is the following:^{[24]} Let D, E, F be the orthogonal projections of P onto BC, CA, AB respectively, and H, K, L be the orthogonal projections of P onto the tangents to the triangle's circumcircle at A, B, C respectively. Then
$PH+PK+PL\geq 2(PD+PE+PF).$
With orthogonal projections H, K, L from P onto the tangents to the triangle's circumcircle at A, B, C respectively, we have^{[25]}
There are various inequalities for an arbitrary interior or exterior point in the plane in terms of the radius r of the triangle's inscribed circle. For example,^{[27]}^{:p. 109}
Let ABC be a triangle, let G be its centroid, and let D, E, and F be the midpoints of BC, CA, and AB, respectively. For any point P in the plane of ABC:
also in terms of the semiperimeter.^{[5]}^{:p. 206}^{[7]}^{:p. 99} Here the expression ${\sqrt {R^{2}-2Rr}}=d$ where d is the distance between the incenter and the circumcenter. In the latter double inequality, the first part holds with equality if and only if the triangle is isosceles with an apex angle of at least 60°, and the last part holds with equality if and only if the triangle is isosceles with an apex angle of at most 60°. Thus both are equalities if and only if the triangle is equilateral.^{[7]}^{:Thm. 1}
where $Q=R^{2}$ if the circumcenter is on or outside of the incircle and $Q=4R^{2}r^{2}\left({\frac {(R-d)^{2}-r^{2}}{(R-d)^{4}}}\right)$ if the circumcenter is inside the incircle. The circumcenter is inside the incircle if and only if^{[32]}
Moreover, for circumcenter O, let lines AO, BO, and CO intersect the opposite sides BC, CA, and AB at U, V, and W respectively. Then^{[2]}^{:p.17,#718}
$OU+OV+OW\geq {\frac {3}{2}}R.$
For an acute triangle the distance between the circumcenter O and the orthocenter H satisfies^{[2]}^{:p.26,#954}
$OH<R,$
with the opposite inequality holding for an obtuse triangle.
The circumradius is at least twice the distance between the first and second Brocard pointsB_{1} and B_{2}:^{[38]}
for circumradius R, and^{[2]}^{:p.181,#264.4}^{[2]}^{:p.45,#1282}
$0\leq (IX-IA)+(IY-IB)+(IZ-IC)\leq 2(R-2r).$
If the incircle is tangent to the sides at D, E, F, then^{[2]}^{:p.115,#2875}
$EF^{2}+FD^{2}+DE^{2}\leq {\frac {s^{2}}{3}}$
for semiperimeter s.
Inscribed figures
Inscribed hexagon
If a tangential hexagon is formed by drawing three segments tangent to a triangle's incircle and parallel to a side, so that the hexagon is inscribed in the triangle with its other three sides coinciding with parts of the triangle's sides, then^{[2]}^{:p.42,#1245}
${\text{Perimeter of hexagon}}\leq {\frac {2}{3}}({\text{Perimeter of triangle}}).$
Inscribed triangle
If three points D, E, F on the respective sides AB, BC, and CA of a reference triangle ABC are the vertices of an inscribed triangle, which thereby partitions the reference triangle into four triangles, then the area of the inscribed triangle is greater than the area of at least one of the other interior triangles, unless the vertices of the inscribed triangle are at the midpoints of the sides of the reference triangle (in which case the inscribed triangle is the medial triangle and all four interior triangles have equal areas):^{[9]}^{:p.137}
An acute triangle has three inscribed squares, each with one side coinciding with part of a side of the triangle and with the square's other two vertices on the remaining two sides of the triangle. (A right triangle has only two distinct inscribed squares.) If one of these squares has side length x_{a} and another has side length x_{b} with x_{a} < x_{b}, then^{[39]}^{:p. 115}
Moreover, for any square inscribed in any triangle we have^{[2]}^{:p.18,#729}^{[39]}
${\frac {\text{Area of triangle}}{\text{Area of inscribed square}}}\geq 2.$
Euler line
A triangle's Euler line goes through its orthocenter, its circumcenter, and its centroid, but does not go through its incenter unless the triangle is isosceles.^{[16]}^{:p.231} For all non-isosceles triangles, the distance d from the incenter to the Euler line satisfies the following inequalities in terms of the triangle's longest medianv, its longest side u, and its semiperimeter s:^{[16]}^{:p. 234,Propos.5}
For all of these ratios, the upper bound of 1/3 is the tightest possible.^{[16]}^{:p.235,Thm.6}
Right triangle
In right triangles the legs a and b and the hypotenusec obey the following, with equality only in the isosceles case:^{[1]}^{:p. 280}
$a+b\leq c{\sqrt {2}}.$
In terms of the inradius, the hypotenuse obeys^{[1]}^{:p. 281}
$2r\leq c({\sqrt {2}}-1),$
and in terms of the altitude from the hypotenuse the legs obey^{[1]}^{:p. 282}
$h_{c}\leq {\frac {\sqrt {2}}{4}}(a+b).$
Isosceles triangle
If the two equal sides of an isosceles triangle have length a and the other side has length c, then the internal angle bisectort from one of the two equal-angled vertices satisfies^{[2]}^{:p.169,#$\eta$44}
For any point P in the plane of an equilateral triangleABC, the distances of P from the vertices, PA, PB, and PC, are such that, unless P is on the triangle's circumcircle, they obey the basic triangle inequality and thus can themselves form the sides of a triangle:^{[1]}^{:p. 279}
$PA+PB>PC,\quad PB+PC>PA,\quad PC+PA>PB.$
However, when P is on the circumcircle the sum of the distances from P to the nearest two vertices exactly equals the distance to the farthest vertex.
A triangle is equilateral if and only if, for every point P in the plane, with distances PD, PE, and PF to the triangle's sides and distances PA, PB, and PC to its vertices,^{[2]}^{:p.178,#235.4}
The hinge theorem or open-mouth theorem states that if two sides of one triangle are congruent to two sides of another triangle, and the included angle of the first is larger than the included angle of the second, then the third side of the first triangle is longer than the third side of the second triangle. That is, in triangles ABC and DEF with sides a, b, c, and d, e, f respectively (with a opposite A etc.), if a = d and b = e and angle C > angle F, then
$c>f.$
The converse also holds: if c > f, then C > F.
The angles in any two triangles ABC and DEF are related in terms of the cotangent function according to^{[6]}
^Chakerian, G. D. "A Distorted View of Geometry." Ch. 7 in Mathematical Plums (R. Honsberger, editor). Washington, DC: Mathematical Association of America, 1979: 147.
^Minda, D., and Phelps, S., "Triangles, ellipses, and cubic polynomials", American Mathematical Monthly 115, October 2008, 679-689: Theorem 4.1.
^Dao Thanh Oai, Nguyen Tien Dung, and Pham Ngoc Mai, "A strengthened version of the Erd?s-Mordell inequality", Forum Geometricorum 16 (2016), pp. 317--321, Theorem 2 http://forumgeom.fau.edu/FG2016volume16/FG201638.pdf
^Dan S ?tefan Marinescu and Mihai Monea, "About a Strengthened Version of the Erdo ?s-Mordell Inequality", Forum Geometricorum Volume 17 (2017), pp. 197-202, Corollary 7. http://forumgeom.fau.edu/FG2017volume17/FG201723.pdf
^ ^{a}^{b}Yurii, N. Maltsev and Anna S. Kuzmina, "An improvement of Birsan's inequalities for the sides of a triangle", Forum Geometricorum 16, 2016, pp. 81-84.
^Blundon, W. J. (1965). "Inequalities associated with the triangle". Canad. Math. Bull. 8 (5): 615-626. doi:10.4153/cmb-1965-044-9.