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 = (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
or equivalently
In addition,
where the value of the right side is the lowest possible bound,Posamentier, Alfred S. and Lehmann, Ingmar. The Secrets of Triangles, Prometheus Books, 2012. 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
If angle C is obtuse (greater than 90°) then
if C is acute (less than 90°) then
The in-between case of equality when C is a right angle is the Pythagorean theorem.
In general,Inequalities proposed in “Crux Mathematicorum” and elsewhere, .
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, thenNyugen, Minh Ha, and Dergiades, Nikolaos. "Garfunkel's Inequality", Forum Geometricorum 4, 2004, 153–156. http://forumgeom.fau.edu/FG2004volume4/FG200419index.html
Equivalently, if constitute the sides of a triangle and the triangle's centroid is inside the incircle then the equation 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:
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
for semi-perimeter s, with equality only in the equilateral case.
Lu, Zhiqin. "An optimal inequality", Mathematical Gazette 91, November 2007, 521–523.
Svrtan, Dragutin and Veljan, Darko. "Non-Euclidean versions of some classical triangle inequalities", Forum Geometricorum 12, 2012, 197–209. http://forumgeom.fau.edu/FG2012volume12/FG201217index.html
where the golden ratio.
Scott, J. A., "A cotangent inequality for two triangles", Mathematical Gazette 89, November 2005, 473–474.
For circumradius R and inradius r we have
with equality if and only if the triangle is isosceles with apex angle greater than or equal to 60°; and
with equality if and only if the triangle is isosceles with apex angle less than or equal to 60°.
We also have
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
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:
If a point D is in the interior of triangle ABC, then
For an acute triangle we have
with the reverse inequality holding for an obtuse triangle.
Furthermore, for non-obtuse triangles we haveShattuck, Mark. “A Geometric Inequality for Cyclic Quadrilaterals”, Forum Geometricorum 18, 2018, 141-154. http://forumgeom.fau.edu/FG2018volume18/FG201822.pdf
with equality if and only if it is a right triangle with hypotenuse AC.
Area
Weitzenböck's inequality is, in terms of area T,
with equality only in the equilateral case. This is a corollary of the Hadwiger–Finsler inequality, which is
Also,
and
From the rightmost upper bound on T, using the arithmetic-geometric mean inequality, is obtained the isoperimetric inequality for triangles:
for semiperimeter s. This is sometimes stated in terms of perimeter p as
with equality for the equilateral triangle.Chakerian, G. D. "A Distorted View of Geometry." Ch. 7 in Mathematical Plums (R. Honsberger, editor). Washington, DC: Mathematical Association of America, 1979: 147. This is strengthened by
Bonnesen's inequality also strengthens the isoperimetric inequality:
We also have
Torrejon, Ricardo M. "On an Erdos inscribed triangle inequality", Forum Geometricorum 5, 2005, 137–141. http://forumgeom.fau.edu/FG2005volume5/FG200519index.html
with equality only in the equilateral case;
for semiperimeter s; and
Ono's inequality for acute triangles (those with all angles less than 90°) is
The area of the triangle can be compared to the area of the incircle:
with equality only for the equilateral triangle.Minda, D., and Phelps, S., "Triangles, ellipses, and cubic polynomials", American Mathematical Monthly 115, October 2008, 679–689: Theorem 4.1.
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
Let the interior angle bisectors of A, B, and C meet the opposite sides at D, E, and F. Then
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.Henry Bottomley, “Medians and Area Bisectors of a Triangle” http://www.se16.info/js/halfarea.htm
Medians and centroid
The three medians of a triangle each connect a vertex with the midpoint of the opposite side, and the sum of their lengths satisfies
Moreover,
with equality only in the equilateral case, and for inradius r,
If we further denote the lengths of the medians extended to their intersections with the circumcircle as Ma , Mb , and Mc , then
The centroid G is the intersection of the medians. Let AG, BG, and CG meet the circumcircle at U, V, and W respectively. Then both
and
in addition,
For an acute triangle we have
in terms of the circumradius R, while the opposite inequality holds for an obtuse triangle.
Denoting as IA, IB, IC the distances of the incenter from the vertices, the following holds:
The three medians of any triangle can form the sides of another triangle:Benyi, A ́rpad, and C ́́urgus, Branko. "Ceva's triangle inequalities", Mathematical Inequalities & Applications 17 (2), 2014, 591-609.
Furthermore,Michel Bataille, “Constructing a Triangle from Two Vertices and the Symmedian Point”, Forum Geometricorum 18 (2018), 129--133.
Altitudes
The altitudes ha, etc. each connect a vertex to the opposite side and are perpendicular to that side. They satisfy both
and
In addition, if then
We also have
For internal angle bisectors ta, tb, tc from vertices A, B, C and circumcenter R and incenter r, we have
The reciprocals of the altitudes of any triangle can themselves form a triangle:Mitchell, Douglas W., "A Heron-type formula for the reciprocal area of a triangle", Mathematical Gazette 89 (November 2005), 494.
Internal angle bisectors and incenter
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 ta etc. satisfy
in terms of the sides, and
in terms of the altitudes and medians, and likewise for tb and tc . Further,
in terms of the medians, and
in terms of the altitudes, inradius r and circumradius R.
Let Ta , Tb , and Tc be the lengths of the angle bisectors extended to the circumcircle. Then
with equality only in the equilateral case, and
for circumradius R and inradius r, again with equality only in the equilateral case. In addition,.
For incenter I (the intersection of the internal angle bisectors),
For midpoints L, M, N of the sides,
For incenter I, centroid G, circumcenter O, nine-point center N, and orthocenter H, we have for non-equilateral triangles the distance inequalitiesFranzsen, William N.. "The distance from the incenter to the Euler line", Forum Geometricorum 11 (2011): 231–236.
and
and we have the angle inequality
In addition,
where v is the longest median.
Three triangles with vertex at the incenter, OIH, GIH, and OGI, are obtuse:
>
Since these triangles have the indicated obtuse angles, we have
and in fact the second of these is equivalent to a result stronger than the first, shown by Euler:L. Euler, "Solutio facilis problematum quorundam geometricorum difficillimorum", Novi Comm. Acad. Scie. Petropolitanae 11 (1765); reprinted in Opera Omnia, serie prima, vol. 26 (A. Speiser, ed.), n. 325, 139–157.
The larger of two angles of a triangle has the shorter internal angle bisector:Altshiller-Court, Nathan. College Geometry. Dover Publications, 2007.
Perpendicular bisectors of sides
These inequalities deal with the lengths pa etc. of the triangle-interior portions of the perpendicular bisectors of sides of the triangle. Denoting the sides so that we haveMitchell, Douglas W. "Perpendicular bisectors of triangle sides", Forum Geometricorum 13, 2013, 53–59: Theorem 4. http://forumgeom.fau.edu/FG2013volume13/FG201307index.html
and
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
and more strongly than the second of these inequalities is: If is the shortest side of the triangle, then
We also have Ptolemy's inequality
for interior point P and likewise for cyclic permutations of the vertices.