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An acute triangle (or acute-angled triangle) is a
triangle A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, an ...
with three acute
angle In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''Side (plane geometry), sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), vertex'' of the angle. Angles formed by two ...
s (less than 90°). An obtuse triangle (or obtuse-angled triangle) is a triangle with one obtuse angle (greater than 90°) and two acute angles. Since a triangle's angles must sum to 180° in
Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small ...
, no Euclidean triangle can have more than one obtuse angle. Acute and obtuse triangles are the two different types of oblique triangles — triangles that are not
right triangle A right triangle (American English) or right-angled triangle (British), or more formally an orthogonal triangle, formerly called a rectangled triangle ( grc, ὀρθόσγωνία, lit=upright angle), is a triangle in which one angle is a right an ...
s because they do not have a 90° angle.


Properties

In all triangles, the
centroid In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the surface of the figure. The same definition extends to any ob ...
—the intersection of the
medians The Medes ( Old Persian: ; Akkadian: , ; Ancient Greek: ; Latin: ) were an ancient Iranian people who spoke the Median language and who inhabited an area known as Media between western and northern Iran. Around the 11th century BC, th ...
, each of which connects a vertex with the midpoint of the opposite side—and the
incenter In geometry, the incenter of a triangle is a triangle center, a point defined for any triangle in a way that is independent of the triangle's placement or scale. The incenter may be equivalently defined as the point where the internal angle bisec ...
—the center of the circle that is internally tangent to all three sides—are in the interior of the triangle. However, while the
orthocenter In geometry, an altitude of a triangle is a line segment through a vertex and perpendicular to (i.e., forming a right angle with) a line containing the base (the side opposite the vertex). This line containing the opposite side is called the '' ...
and the
circumcenter In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polyg ...
are in an acute triangle's interior, they are exterior to an obtuse triangle. The orthocenter is the intersection point of the triangle's three
altitudes Altitude or height (also sometimes known as depth) is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. The exact definition and reference datum varies according to the context ...
, each of which
perpendicular In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It can ...
ly connects a side to the opposite
vertex Vertex, vertices or vertexes may refer to: Science and technology Mathematics and computer science *Vertex (geometry), a point where two or more curves, lines, or edges meet *Vertex (computer graphics), a data structure that describes the position ...
. In the case of an acute triangle, all three of these segments lie entirely in the triangle's interior, and so they intersect in the interior. But for an obtuse triangle, the altitudes from the two acute angles intersect only the
extensions Extension, extend or extended may refer to: Mathematics Logic or set theory * Axiom of extensionality * Extensible cardinal * Extension (model theory) * Extension (predicate logic), the set of tuples of values that satisfy the predicate * Ex ...
of the opposite sides. These altitudes fall entirely outside the triangle, resulting in their intersection with each other (and hence with the extended altitude from the obtuse-angled vertex) occurring in the triangle's exterior. Likewise, a triangle's circumcenter—the intersection of the three sides' perpendicular bisectors, which is the center of the circle that passes through all three vertices—falls inside an acute triangle but outside an obtuse triangle. The
right triangle A right triangle (American English) or right-angled triangle (British), or more formally an orthogonal triangle, formerly called a rectangled triangle ( grc, ὀρθόσγωνία, lit=upright angle), is a triangle in which one angle is a right an ...
is the in-between case: both its circumcenter and its orthocenter lie on its boundary. In any triangle, any two angle measures ''A'' and ''B'' opposite sides ''a'' and ''b'' respectively are related according to Posamentier, Alfred S. and Lehmann, Ingmar. ''
The Secrets of Triangles ''The Secrets of Triangles: A Mathematical Journey'' is a popular mathematics book on the geometry of triangles. It was written by Alfred S. Posamentier and , and published in 2012 by Prometheus Books. Topics The book consists of ten chapters, ...
'', Prometheus Books, 2012.
:A>B \quad \text \quad a > b. This implies that the longest side in an obtuse triangle is the one opposite the obtuse-angled vertex. 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. (In a right triangle two of these are merged into the same square, so there are only two distinct inscribed squares.) However, an obtuse triangle has only one inscribed square, one of whose sides coincides with part of the longest side of the triangle.Oxman, Victor, and Stupel, Moshe. "Why are the side lengths of the squares inscribed in a triangle so close to each other?" ''Forum Geometricorum'' 13, 2013, 113–115. http://forumgeom.fau.edu/FG2013volume13/FG201311index.html All triangles in which the
Euler line In geometry, the Euler line, named after Leonhard Euler (), is a line determined from any triangle that is not equilateral. It is a central line of the triangle, and it passes through several important points determined from the triangle, inclu ...
is parallel to one side are acute.Wladimir G. Boskoff, Laurent¸iu Homentcovschi, and Bogdan D. Suceava, "Gossard’s Perspector and Projective Consequences", ''Forum Geometricorum'', Volume 13 (2013), 169–184

/ref> This property holds for side BC
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondi ...
(\tan B)(\tan C)=3.


Inequalities


Sides

If angle ''C'' is obtuse then for sides ''a'', ''b'', and ''c'' we have''Inequalities proposed in “
Crux Mathematicorum ''Crux Mathematicorum'' is a scientific journal of mathematics published by the Canadian Mathematical Society. It contains mathematical problems for secondary school and undergraduate students. , its editor-in-chief is Kseniya Garaschuk. The journ ...
”''

:\frac < a^2+b^2 < c^2, with the left inequality approaching equality in the limit only as the apex angle of an isosceles triangle approaches 180°, and with the right inequality approaching equality only as the obtuse angle approaches 90°. If the triangle is acute then :a^2+b^2 > c^2, \quad b^2+c^2 > a^2, \quad c^2+a^2 > b^2.


Altitude

If C is the greatest angle and ''h''''c'' is the altitude from vertex ''C'', then for an acute triangle :\frac < \frac+\frac, with the opposite inequality if C is obtuse.


Medians

With longest side ''c'' and
medians The Medes ( Old Persian: ; Akkadian: , ; Ancient Greek: ; Latin: ) were an ancient Iranian people who spoke the Median language and who inhabited an area known as Media between western and northern Iran. Around the 11th century BC, th ...
''m''''a'' and ''m''''b'' from the other sides, :4c^2 +9a^2b^2 > 16m_a^2m_b^2 for an acute triangle but with the inequality reversed for an obtuse triangle. The median ''m''''c'' from the longest side is greater or less than the circumradius for an acute or obtuse triangle respectively: :m_c > R for acute triangles, with the opposite for obtuse triangles.


Area

Ono's inequality In mathematics, Ono's inequality is a theorem about triangles in the Euclidean plane. In its original form, as conjectured by T. Ono in 1914, the inequality is actually false; however, the statement is true for acute triangles and right triangles, ...
for the area ''A'', :27 (b^2 + c^2 - a^2)^2 (c^2 + a^2 - b^2)^2 (a^2 + b^2 - c^2)^2 \leq (4 A)^6, holds for all acute triangles but not for all obtuse triangles.


Trigonometric functions

For an acute triangle we have, for angles ''A'', ''B'', and ''C'', :\cos^2A+\cos^2B+\cos^2C < 1, with the reverse inequality holding for an obtuse triangle. For an acute triangle with circumradius ''R'', :a\cos^3 A +b\cos^3 B +c\cos^3 C \leq \frac and :\cos^3A+\cos^3B+\cos^3C+\cos A\cos B\cos C\geq\frac. For an acute triangle, :\sin^2 A+\sin^2 B+\sin^2 C > 2, with the reverse inequality for an obtuse triangle. For an acute triangle, :\sin A \cdot \sin B +\sin B \cdot \sin C + \sin C \cdot \sin A \leq (\cos A+\cos B+\cos C)^2. For any triangle the
triple tangent identity In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involvin ...
states that the sum of the angles'
tangents In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...
equals their product. Since an acute angle has a positive tangent value while an obtuse angle has a negative one, the expression for the product of the tangents shows that :\tan A+\tan B+\tan C = \tan A \cdot \tan B \cdot \tan C > 0 for acute triangles, while the opposite direction of inequality holds for obtuse triangles. We have :\tan A +\tan B+\tan C \geq 2(\sin 2A+\sin 2B+\sin 2C) for acute triangles, and the reverse for obtuse triangles. For all acute triangles, :(\tan A+\tan B+\tan C)^2 \geq (\sec A+1)^2+(\sec B+1)^2+(\sec C+1)^2. For all acute triangles with
inradius In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. ...
''r'' and
circumradius In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polyg ...
''R'', :a\tan A+ b\tan B+c\tan C \geq 10R-2r. For an acute triangle with area ''K'', :(\sqrt+\sqrt+\sqrt)^2 \leq \frac.


Circumradius, inradius, and exradii

In an acute triangle, the sum of the circumradius ''R'' and the inradius ''r'' is less than half the sum of the shortest sides ''a'' and ''b'': :R+r < \frac, while the reverse inequality holds for an obtuse triangle. For an acute triangle with
medians The Medes ( Old Persian: ; Akkadian: , ; Ancient Greek: ; Latin: ) were an ancient Iranian people who spoke the Median language and who inhabited an area known as Media between western and northern Iran. Around the 11th century BC, th ...
''m''''a'' , ''m''''b'' , and ''m''''c'' and circumradius ''R'', we have :m_a^2+m_b^2+m_c^2 > 6R^2 while the opposite inequality holds for an obtuse triangle. Also, an acute triangle satisfies :r^2+r_a^2+r_b^2+r_c^2 < 8R^2, in terms of the
excircle In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. ...
radii ''r''''a'' , ''r''''b'' , and ''r''''c'' , again with the reverse inequality holding for an obtuse triangle. For an acute triangle with semiperimeter ''s'', :s-r >2R, and the reverse inequality holds for an obtuse triangle. For an acute triangle with area ''K'', :ab+bc+ca \geq 2R(R+r)+\frac.


Distances involving triangle centers

For an acute triangle the distance between the circumcenter ''O'' and the orthocenter ''H'' satisfies :OH < R, with the opposite inequality holding for an obtuse triangle. For an acute triangle the distance between the incircle center ''I'' and orthocenter ''H'' satisfies :IH < r\sqrt, where ''r'' is the
inradius In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. ...
, with the reverse inequality for an obtuse triangle.


Inscribed square

If one of the inscribed squares of an acute triangle has side length ''x''''a'' and another has side length ''x''''b'' with ''x''''a'' < ''x''''b'', then :1 \geq \frac \geq \frac \approx 0.94.


Two triangles

If two obtuse triangles have sides (''a, b, c'') and (''p, q, r'') with ''c'' and ''r'' being the respective longest sides, then :ap+bq < cr.


Examples


Triangles with special names

The
Calabi triangle The Calabi triangle is a special triangle found by Eugenio Calabi and defined by its property of having three different placements for the largest square that it contains. It is an obtuse isosceles triangle with an irrational but algebraic ratio ...
, which is the only non-equilateral triangle for which the largest square that fits in the interior can be positioned in any of three different ways, is obtuse and isosceles with base angles 39.1320261...° and third angle 101.7359477...°. The
equilateral triangle In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each othe ...
, with three 60° angles, is acute. The
Morley triangle In plane geometry, Morley's trisector theorem states that in any triangle, the three points of intersection of the adjacent angle trisectors form an equilateral triangle, called the first Morley triangle or simply the Morley triangle. The theorem ...
, formed from any triangle by the intersections of its adjacent angle trisectors, is equilateral and hence acute. The
golden triangle Golden Triangle may refer to: Places Asia * Golden Triangle (Southeast Asia), named for its opium production * Golden Triangle (Yangtze), China, named for its rapid economic development * Golden Triangle (India), comprising the popular tourist ...
is the
isosceles triangle In geometry, an isosceles triangle () is a triangle that has two sides of equal length. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at least'' two sides of equal length, the latter versio ...
in which the ratio of the duplicated side to the base side equals the
golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ( ...
. It is acute, with angles 36°, 72°, and 72°, making it the only triangle with angles in the proportions 1:2:2. The
heptagonal triangle A heptagonal triangle is an obtuse scalene triangle whose vertices coincide with the first, second, and fourth vertices of a regular heptagon (from an arbitrary starting vertex). Thus its sides coincide with one side and the adjacent shorter an ...
, with sides coinciding with a side, the shorter diagonal, and the longer diagonal of a regular
heptagon In geometry, a heptagon or septagon is a seven-sided polygon or 7-gon. The heptagon is sometimes referred to as the septagon, using "sept-" (an elision of ''septua-'', a Latin-derived numerical prefix, rather than ''hepta-'', a Greek-derived num ...
, is obtuse, with angles \pi/7, 2\pi/7, and 4\pi/7.


Triangles with integer sides

The only triangle with consecutive integers for an altitude and the sides is acute, having sides (13,14,15) and altitude from side 14 equal to 12. The smallest-perimeter triangle with integer sides in arithmetic progression, and the smallest-perimeter integer-sided triangle with distinct sides, is obtuse: namely the one with sides (2, 3, 4). The only triangles with one angle being twice another and having integer sides in
arithmetic progression An arithmetic progression or arithmetic sequence () is a sequence of numbers such that the difference between the consecutive terms is constant. For instance, the sequence 5, 7, 9, 11, 13, 15, . . . is an arithmetic progression with a common differ ...
are acute: namely, the (4,5,6) triangle and its multiples.Mitchell, Douglas W., "The 2:3:4, 3:4:5, 4:5:6, and 3:5:7 triangles," ''Mathematical Gazette'' 92, July 2008. There are no acute integer-sided triangles with
area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape A shape or figure is a graphics, graphical representation of an obje ...
=
perimeter A perimeter is a closed path that encompasses, surrounds, or outlines either a two dimensional shape or a one-dimensional length. The perimeter of a circle or an ellipse is called its circumference. Calculating the perimeter has several pract ...
, but there are three obtuse ones, having sides
L. E. Dickson Leonard Eugene Dickson (January 22, 1874 – January 17, 1954) was an American mathematician. He was one of the first American researchers in abstract algebra, in particular the theory of finite fields and classical groups, and is also reme ...
, ''
History of the Theory of Numbers ''History of the Theory of Numbers'' is a three-volume work by L. E. Dickson summarizing work in number theory up to about 1920. The style is unusual in that Dickson mostly just lists results by various authors, with little further discussion. T ...
, vol.2'', 181.
(6,25,29), (7,15,20), and (9,10,17). The smallest integer-sided triangle with three rational
medians The Medes ( Old Persian: ; Akkadian: , ; Ancient Greek: ; Latin: ) were an ancient Iranian people who spoke the Median language and who inhabited an area known as Media between western and northern Iran. Around the 11th century BC, th ...
is acute, with sidesSierpiński, Wacław. ''
Pythagorean Triangles ''Pythagorean Triangles'' is a book on right triangles, the Pythagorean theorem, and Pythagorean triples. It was originally written in the Polish language by Wacław Sierpiński (titled ''Trójkąty pitagorejskie''), and published in Warsaw in 1 ...
'', Dover Publ., 2003 (orig. 1962).
(68, 85, 87).
Heron triangle In geometry, a Heronian triangle (or Heron triangle) is a triangle whose side lengths , , and and area are all integers. Heronian triangles are named after Heron of Alexandria, based on their relation to Heron's formula. Heron's formula implies ...
s have integer sides and integer area. The oblique Heron triangle with the smallest perimeter is acute, with sides (6, 5, 5). The two oblique Heron triangles that share the smallest area are the acute one with sides (6, 5, 5) and the obtuse one with sides (8, 5, 5), the area of each being 12.


References

* * {{Polygons Types of triangles