Triangle

Ancient Greek mathematician Euclid defined three types of triangle according to the lengths of their sides:[1][2]

Greek: τῶν δὲ τριπλεύρων σχημάτων ἰσόπλευρον μὲν τρίγωνόν ἐστι τὸ τὰς τρεῖς ἴσας ἔχον πλευράς, ἰσοσκελὲς δὲ τὸ τὰς δύο μόνας ἴσας ἔχον πλευράς, σκαληνὸν δὲ τὸ τὰς τρεῖς ἀνίσους ἔχον πλευράς, lit. 'Of trilateral figures, an isopleuron [equilateral] triangle is that which has its three sides equal, an isosceles that which has two of its sides alone equal, and a scalene that which has its three sides unequal.'[3]

• An equilateral triangle (Greek: ἰσόπλευρον, romanized: isópleuron, lit. 'equal sides') has three sides of the same length. An equilateral triangle is also a regular polygon with all angles measuring 60°.[4]
• An isosceles triangle (Greek: ἰσοσκελὲς, romanized: isoskelés, lit. 'equal legs') has two sides of equal length.[note 1][5] An isosceles triangle also has two angles of the same measure, namely the angles opposite to the two sides of the same length. This fact is the content of the isosceles triangle theorem, which was known by Euclid. Some mathematicians define an isosceles triangle to have exactly two equal sides, whereas others define an isosceles triangle as one with at least two equal sides.[5] The latter definition would make all equilateral triangles isosceles triangles. The 45–45–90 right triangle, which appears in the tetrakis square tiling, is isosceles.
• A scalene triangle (Greek: σκαληνὸν, romanized: skalinón, lit. 'unequal') has all its sides of different lengths.[6] Equivalently, it has all angles of different measure.

• 
  Equilateral Triangle
• 
  Isosceles triangle
• 
  Scalene triangle

Hatch marks, also called tick marks, are used in diagrams of triangles and other geometric figures to identify sides of equal lengths. A side can be marked with a pattern of "ticks", short line segments in the form of tally marks; two sides have equal lengths if they are both marked with the same pattern. In a triangle, the pattern is usually no more than 3 ticks. An equilateral triangle has the same pattern on all 3 sides, an isosceles triangle has the same pattern on just 2 sides, and a scalene triangle has different patterns on all sides since no sides are equal.

Similarly, patterns of 1, 2, or 3 concentric arcs inside the angles are used to indicate equal angles: an equilateral triangle has the same pattern on all 3 angles, an isosceles triangle has the same pattern on just 2 angles, and a scalene triangle has different patterns on all angles, since no angles are equal.

The first page of Euclid's Elements, from the world's first printed version (1482), showing the "definitions" section of Book I. The right triangle is labeled "orthogonius", and the two angles shown are "acutus" and "angulus obtusus".

Triangles can also be classified according to their internal angles, measured here in degrees.

• A right triangle (or right-angled triangle) has one of its interior angles measuring 90° (a right angle). The side opposite to the right angle is the hypotenuse, the longest side of the triangle. The other two sides are called the legs or catheti[7] (singular: cathetus) of the triangle. Right triangles obey the Pythagorean theorem: the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse: a² + b² = c², where a and b are the lengths of the legs and c is the length of the hypotenuse. Special right triangles are right triangles with additional properties that make calculations involving them easier. One of the two most famous is the 3–4–5 right triangle, where 3² + 4² = 5². The 3–4–5 triangle is also known as the Egyptian triangle.[8] In this situation, 3, 4, and 5 are a Pythagorean triple. The other one is an isosceles triangle that has 2 angles measuring 45 degrees (45–45–90 triangle). Right triangles are fundamental in trigonometry, as they can be used to define trigonometric functions through trigonometric ratios.
  • Triangles that do not have an angle measuring 90° are called oblique triangles.
• A triangle with all interior angles measuring less than 90° is an acute triangle or acute-angled triangle.[2] If c is the length of the longest side, then a² + b² > c², where a and b are the lengths of the other sides.
• A triangle with one interior angle measuring more than 90° is an obtuse triangle or obtuse-angled triangle.[2] If c is the length of the longest side, then a² + b² < c², where a and b are the lengths of the other sides.
• A triangle with an interior angle of 180° (and collinear vertices) is degenerate. A right degenerate triangle has collinear vertices, two of which are coincident.

A triangle that has two angles with the same measure also has two sides with the same length, and therefore it is an isosceles triangle. It follows that in a triangle where all angles have the same measure, all three sides have the same length, and therefore is equilateral.

Right	Obtuse	Acute
	${\displaystyle \quad \underbrace {\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad } _{}}$
	Oblique

Two triangles are said to be similar, if every angle of one triangle has the same measure as the corresponding angle in the other triangle. The corresponding sides of similar triangles have lengths that are in the same proportion, and this property is also sufficient to establish similarity.

Some basic theorems about similar triangles are:

• If and only if one pair of internal angles of two triangles have the same measure as each other, and another pair also have the same measure as each other, the triangles are similar.
• If and only if one pair of corresponding sides of two triangles are in the same proportion as are another pair of corresponding sides, and their included angles have the same measure, then the triangles are similar. (The included angle for any two sides of a polygon is the internal angle between those two sides.)
• If and only if three pairs of corresponding sides of two triangles are all in the same proportion, then the triangles are similar.[note 3]

Two triangles that are congruent have exactly the same size and shape:[note 4] all pairs of corresponding interior angles are equal in measure, and all pairs of corresponding sides have the same length. (This is a total of six equalities, but three are often sufficient to prove congruence.)

Some individually necessary and sufficient conditions for a pair of triangles to be congruent are:

• SAS Postulate: Two sides in a triangle have the same length as two sides in the other triangle, and the included angles have the same measure.
• ASA: Two interior angles and the included side in a triangle have the same measure and length, respectively, as those in the other triangle. (The included side for a pair of angles is the side that is common to them.)
• SSS: Each side of a triangle has the same length as a corresponding side of the other triangle.
• AAS: Two angles and a corresponding (non-included) side in a triangle have the same measure and length, respectively, as those in the other triangle. (This is sometimes referred to as AAcorrS and then includes ASA above.)

Some individually sufficient conditions are:

• Hypotenuse-Leg (HL) Theorem: The hypotenuse and a leg in a right triangle have the same length as those in another right triangle. This is also called RHS (right-angle, hypotenuse, side).
• Hypotenuse-Angle Theorem: The hypotenuse and an acute angle in one right triangle have the same length and measure, respectively, as those in the other right triangle. This is just a particular case of the AAS theorem.

An important condition is:

• Side-Side-Angle (or Angle-Side-Side) condition: If two sides and a corresponding non-included angle of a triangle have the same length and measure, respectively, as those in another triangle, then this is not sufficient to prove congruence; but if the angle given is opposite to the longer side of the two sides, then the triangles are congruent. The Hypotenuse-Leg Theorem is a particular case of this criterion. The Side-Side-Angle condition does not by itself guarantee that the triangles are congruent because one triangle could be obtuse-angled and the other acute-angled.

Two right triangles are similar if and only if they have an acute angle of the same measure. It follows that the six possible ratios between side lengths of a right triangle depend only of the measure of one of the acute angles. These ratios are called trigonometric ratios and are commonly used as a definition of trigonometric functions.

The Pythagorean theorem

A central theorem is the Pythagorean theorem, which states in any right triangle, the square of the length of the hypotenuse equals the sum of the squares of the lengths of the two other sides. If the hypotenuse has length c, and the legs have lengths a and b, then the theorem states that

${\displaystyle a^{2}+b^{2}=c^{2}.}$

The converse is true: if the lengths of the sides of a triangle satisfy the above equation, then the triangle has a right angle opposite side c.

Some other facts about right triangles:

• The acute angles of a right triangle are complementary.

${\displaystyle a+b+90^{\circ }=180^{\circ }\Rightarrow a+b=90^{\circ }\Rightarrow a=90^{\circ }-b.}$

• If the legs of a right triangle have the same length, then the angles opposite those legs have the same measure. Since these angles are complementary, it follows that each measures 45 degrees. By the Pythagorean theorem, the length of the hypotenuse is the length of a leg times √2.
• In a right triangle with acute angles measuring 30 and 60 degrees, the hypotenuse is twice the length of the shorter side, and the longer side is equal to the length of the shorter side times √3:

${\displaystyle c=2a\,}$

${\displaystyle b=a\times {\sqrt {3}}.}$

For all triangles, angles and sides are related by the law of cosines and law of sines (also called the cosine rule and sine rule).

The triangle inequality states that the sum of the lengths of any two sides of a triangle must be greater than or equal to the length of the third side. That sum can equal the length of the third side only in the case of a degenerate triangle, one with collinear vertices. It is not possible for that sum to be less than the length of the third side. A triangle with three given positive side lengths exists if and only if those side lengths satisfy the triangle inequality.

Three given angles form a non-degenerate triangle (and indeed an infinitude of them) if and only if both of these conditions hold: (a) each of the angles is positive, and (b) the angles sum to 180°. If degenerate triangles are permitted, angles of 0° are permitted.

Three positive angles α, β, and γ, each of them less than 180°, are the angles of a triangle if and only if any one of the following conditions holds:

${\displaystyle \tan {\frac {\alpha }{2}}\tan {\frac {\beta }{2}}+\tan {\frac {\beta }{2}}\tan {\frac {\gamma }{2}}+\tan {\frac {\gamma }{2}}\tan {\frac {\alpha }{2}}=1,}$[10]

${\displaystyle \sin ^{2}{\frac {\alpha }{2}}+\sin ^{2}{\frac {\beta }{2}}+\sin ^{2}{\frac {\gamma }{2}}+2\sin {\frac {\alpha }{2}}\sin {\frac {\beta }{2}}\sin {\frac {\gamma }{2}}=1,}$[10]

${\displaystyle \sin(2\alpha )+\sin(2\beta )+\sin(2\gamma )=4\sin(\alpha )\sin(\beta )\sin(\gamma ),}$

${\displaystyle \cos ^{2}\alpha +\cos ^{2}\beta +\cos ^{2}\gamma +2\cos(\alpha )\cos(\beta )\cos(\gamma )=1,}$[11]

${\displaystyle \tan(\alpha )+\tan(\beta )+\tan(\gamma )=\tan(\alpha )\tan(\beta )\tan(\gamma ),}$

the last equality applying only if none of the angles is 90° (so the tangent function's value is always finite).

A right triangle always includes a 90° (π/2 radians) angle, here with label C. Angles A and B may vary. Trigonometric functions specify the relationships among side lengths and interior angles of a right triangle.

In right triangles, the trigonometric ratios of sine, cosine and tangent can be used to find unknown angles and the lengths of unknown sides. The sides of the triangle are known as follows:

• The hypotenuse is the side opposite the right angle, or defined as the longest side of a right-angled triangle, in this case h.
• The opposite side is the side opposite to the angle we are interested in, in this case a.
• The adjacent side is the side that is in contact with the angle we are interested in and the right angle, hence its name. In this case the adjacent side is b.

The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. In our case

${\displaystyle \sin A={\frac {\text{opposite side}}{\text{hypotenuse}}}={\frac {a}{h}}\,.}$

This ratio does not depend on the particular right triangle chosen, as long as it contains the angle A, since all those triangles are similar.

The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse. In our case

${\displaystyle \cos A={\frac {\text{adjacent side}}{\text{hypotenuse}}}={\frac {b}{h}}\,.}$

The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side. In our case

${\displaystyle \tan A={\frac {\text{opposite side}}{\text{adjacent side}}}={\frac {a}{b}}={\frac {\sin A}{\cos A}}\,.}$

The acronym "SOH-CAH-TOA" is a useful mnemonic for these ratios.

The inverse trigonometric functions can be used to calculate the internal angles for a right angled triangle with the length of any two sides.

Arcsin can be used to calculate an angle from the length of the opposite side and the length of the hypotenuse.

${\displaystyle \theta =\arcsin \left({\frac {\text{opposite side}}{\text{hypotenuse}}}\right)}$

Arccos can be used to calculate an angle from the length of the adjacent side and the length of the hypotenuse.

${\displaystyle \theta =\arccos \left({\frac {\text{adjacent side}}{\text{hypotenuse}}}\right)}$

Arctan can be used to calculate an angle from the length of the opposite side and the length of the adjacent side.

${\displaystyle \theta =\arctan \left({\frac {\text{opposite side}}{\text{adjacent side}}}\right)}$

In introductory geometry and trigonometry courses, the notation sin⁻¹, cos⁻¹, etc., are often used in place of arcsin, arccos, etc. However, the arcsin, arccos, etc., notation is standard in higher mathematics where trigonometric functions are commonly raised to powers, as this avoids confusion between multiplicative inverse and compositional inverse.

A triangle with sides of length a, b and c and angles of α, β and γ respectively.

The law of sines, or sine rule,[12] states that the ratio of the length of a side to the sine of its corresponding opposite angle is constant, that is

${\displaystyle {\frac {a}{\sin \alpha }}={\frac {b}{\sin \beta }}={\frac {c}{\sin \gamma }}=2R}$, where R is the radius of the circumscribed circle of the given triangle.

Another interpretation of this theorem is that every triangle with angles α, β and γ is similar to a triangle with side lengths equal to sin α, sin β and sin γ. This triangle can be constructed by first constructing a circle of diameter 1, and inscribing in it two of the angles of the triangle. The length of the sides of that triangle will be sin α, sin β and sin γ. The side whose length is sin α is opposite to the angle whose measure is α, etc.

The law of cosines, or cosine rule, connects the length of an unknown side of a triangle to the length of the other sides and the angle opposite to the unknown side.[12] As per the law:

For a triangle with length of sides a, b, c and angles of α, β, γ respectively, given two known lengths of a triangle a and b, and the angle between the two known sides γ (or the angle opposite to the unknown side c), to calculate the third side c, the following formula can be used:

${\displaystyle c^{2}\ =a^{2}+b^{2}-2ab\cos(\gamma )}$

${\displaystyle b^{2}\ =a^{2}+c^{2}-2ac\cos(\beta )}$

${\displaystyle a^{2}\ =b^{2}+c^{2}-2bc\cos(\alpha )}$

If the lengths of all three sides of any triangle are known the three angles can be calculated:

${\displaystyle \alpha =\arccos \left({\frac {b^{2}+c^{2}-a^{2}}{2bc}}\right)}$

${\displaystyle \beta =\arccos \left({\frac {a^{2}+c^{2}-b^{2}}{2ac}}\right)}$

${\displaystyle \gamma =\arccos \left({\frac {a^{2}+b^{2}-c^{2}}{2ab}}\right)}$

The law of tangents, or tangent rule, can be used to find a side or an angle when two sides and an angle or two angles and a side are known. It states that:[13]

${\displaystyle {\frac {a-b}{a+b}}={\frac {\tan[{\frac {1}{2}}(\alpha -\beta )]}{\tan[{\frac {1}{2}}(\alpha +\beta )]}}.}$

"Solution of triangles" is the main trigonometric problem: to find missing characteristics of a triangle (three angles, the lengths of the three sides etc.) when at least three of these characteristics are given. The triangle can be located on a plane or on a sphere. This problem often occurs in various trigonometric applications, such as geodesy, astronomy, construction, navigation etc.

The medians and the sides are related by[17]^: p.70

${\displaystyle {\frac {3}{4}}(a^{2}+b^{2}+c^{2})=m_{a}^{2}+m_{b}^{2}+m_{c}^{2}}$

and

${\displaystyle m_{a}={\frac {1}{2}}{\sqrt {2b^{2}+2c^{2}-a^{2}}}={\sqrt {{\frac {1}{2}}(a^{2}+b^{2}+c^{2})-{\frac {3}{4}}a^{2}}}}$,

and equivalently for m_b and m_c.

For angle A opposite side a, the length of the internal angle bisector is given by[18]

${\displaystyle w_{A}={\frac {2{\sqrt {bcs(s-a)}}}{b+c}}={\sqrt {bc\left[1-{\frac {a^{2}}{(b+c)^{2}}}\right]}}={\frac {2bc}{b+c}}\cos {\frac {A}{2}},}$

for semiperimeter s, where the bisector length is measured from the vertex to where it meets the opposite side.

The interior perpendicular bisectors are given by

${\displaystyle p_{a}={\frac {2aT}{a^{2}+b^{2}-c^{2}}},}$

${\displaystyle p_{b}={\frac {2bT}{a^{2}+b^{2}-c^{2}}},}$

${\displaystyle p_{c}={\frac {2cT}{a^{2}-b^{2}+c^{2}}},}$

where the sides are ${\displaystyle a\geq b\geq c}$ and the area is ${\displaystyle T.}$[19]^{: Thm 2}

The altitude from, for example, the side of length a is

${\displaystyle h_{a}={\frac {2T}{a}}.}$

The following formulas involve the circumradius R and the inradius r:

${\displaystyle R={\sqrt {\frac {a^{2}b^{2}c^{2}}{(a+b+c)(-a+b+c)(a-b+c)(a+b-c)}}};}$

${\displaystyle r={\sqrt {\frac {(-a+b+c)(a-b+c)(a+b-c)}{4(a+b+c)}}};}$

${\displaystyle {\frac {1}{r}}={\frac {1}{h_{a}}}+{\frac {1}{h_{b}}}+{\frac {1}{h_{c}}}}$

where h_a etc. are the altitudes to the subscripted sides;[17]^: p.79

${\displaystyle {\frac {r}{R}}={\frac {4T^{2}}{sabc}}=\cos \alpha +\cos \beta +\cos \gamma -1;}$[11]

and

${\displaystyle 2Rr={\frac {abc}{a+b+c}}}$.

The product of two sides of a triangle equals the altitude to the third side times the diameter D of the circumcircle:[17]^: p.64

${\displaystyle ab=h_{c}D,\quad \quad bc=h_{a}D,\quad ca=h_{b}D.}$

Suppose two adjacent but non-overlapping triangles share the same side of length f and share the same circumcircle, so that the side of length f is a chord of the circumcircle and the triangles have side lengths (a, b, f) and (c, d, f), with the two triangles together forming a cyclic quadrilateral with side lengths in sequence (a, b, c, d). Then[20]^: 84

${\displaystyle f^{2}={\frac {(ac+bd)(ad+bc)}{(ab+cd)}}.\,}$

Let G be the centroid of a triangle with vertices A, B, and C, and let P be any interior point. Then the distances between the points are related by[20]^: 174

${\displaystyle (PA)^{2}+(PB)^{2}+(PC)^{2}=(GA)^{2}+(GB)^{2}+(GC)^{2}+3(PG)^{2}.\,}$

The sum of the squares of the triangle's sides equals three times the sum of the squared distances of the centroid from the vertices:

${\displaystyle AB^{2}+BC^{2}+CA^{2}=3(GA^{2}+GB^{2}+GC^{2}).}$[21]

Let q_a, q_b, and q_c be the distances from the centroid to the sides of lengths a, b, and c. Then[20]^: 173

${\displaystyle {\frac {q_{a}}{q_{b}}}={\frac {b}{a}},\quad \quad {\frac {q_{b}}{q_{c}}}={\frac {c}{b}},\quad \quad {\frac {q_{a}}{q_{c}}}={\frac {c}{a}}\,}$

and

${\displaystyle q_{a}\cdot a=q_{b}\cdot b=q_{c}\cdot c={\frac {2}{3}}T\,}$

for area T.

Carnot's theorem states that the sum of the distances from the circumcenter to the three sides equals the sum of the circumradius and the inradius.[17]^: p.83 Here a segment's length is considered to be negative if and only if the segment lies entirely outside the triangle. This method is especially useful for deducing the properties of more abstract forms of triangles, such as the ones induced by Lie algebras, that otherwise have the same properties as usual triangles.

Euler's theorem states that the distance d between the circumcenter and the incenter is given by[17]^: p.85

${\displaystyle \displaystyle d^{2}=R(R-2r)}$

or equivalently

${\displaystyle {\frac {1}{R-d}}+{\frac {1}{R+d}}={\frac {1}{r}},}$

where R is the circumradius and r is the inradius. Thus for all triangles R ≥ 2r, with equality holding for equilateral triangles.

If we denote that the orthocenter divides one altitude into segments of lengths u and v, another altitude into segment lengths w and x, and the third altitude into segment lengths y and z, then uv = wx = yz.[17]^: p.94

The distance from a side to the circumcenter equals half the distance from the opposite vertex to the orthocenter.[17]^: p.99

The sum of the squares of the distances from the vertices to the orthocenter H plus the sum of the squares of the sides equals twelve times the square of the circumradius:[17]^: p.102

${\displaystyle AH^{2}+BH^{2}+CH^{2}+a^{2}+b^{2}+c^{2}=12R^{2}.}$

In addition to the law of sines, the law of cosines, the law of tangents, and the trigonometric existence conditions given earlier, for any triangle

${\displaystyle a=b\cos C+c\cos B,\quad b=c\cos A+a\cos C,\quad c=a\cos B+b\cos A.}$

The Morley triangle, resulting from the trisection of each interior angle. This is an example of a finite subdivision rule.

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 Morley triangle.

As discussed above, every triangle has a unique inscribed circle (incircle) that is interior to the triangle and tangent to all three sides.

Every triangle has a unique Steiner inellipse which is interior to the triangle and tangent at the midpoints of the sides. Marden's theorem shows how to find the foci of this ellipse.[22] This ellipse has the greatest area of any ellipse tangent to all three sides of the triangle.

The Mandart inellipse of a triangle is the ellipse inscribed within the triangle tangent to its sides at the contact points of its excircles.

For any ellipse inscribed in a triangle ABC, let the foci be P and Q. Then[23]

${\displaystyle {\frac {{\overline {PA}}\cdot {\overline {QA}}}{{\overline {CA}}\cdot {\overline {AB}}}}+{\frac {{\overline {PB}}\cdot {\overline {QB}}}{{\overline {AB}}\cdot {\overline {BC}}}}+{\frac {{\overline {PC}}\cdot {\overline {QC}}}{{\overline {BC}}\cdot {\overline {CA}}}}=1.}$

Every convex polygon with area T can be inscribed in a triangle of area at most equal to 2T. Equality holds (exclusively) for a parallelogram.[24]

The Lemoine hexagon is a cyclic hexagon with vertices given by the six intersections of the sides of a triangle with the three lines that are parallel to the sides and that pass through its symmedian point. In either its simple form or its self-intersecting form, the Lemoine hexagon is interior to the triangle with two vertices on each side of the triangle.

Every acute triangle has three inscribed squares (squares in its interior such that all four of a square's vertices lie on a side of the triangle, so two of them lie on the same side and hence one side of the square coincides with part of a side of the triangle). In a right triangle two of the squares coincide and have a vertex at the triangle's right angle, so a right triangle has only two distinct inscribed squares. An obtuse triangle has only one inscribed square, with a side coinciding with part of the triangle's longest side. Within a given triangle, a longer common side is associated with a smaller inscribed square. If an inscribed square has side of length q_a and the triangle has a side of length a, part of which side coincides with a side of the square, then q_a, a, the altitude h_a from the side a, and the triangle's area T are related according to[25][26]

${\displaystyle q_{a}={\frac {2Ta}{a^{2}+2T}}={\frac {ah_{a}}{a+h_{a}}}.}$

The largest possible ratio of the area of the inscribed square to the area of the triangle is 1/2, which occurs when a² = 2T, q = a/2, and the altitude of the triangle from the base of length a is equal to a. The smallest possible ratio of the side of one inscribed square to the side of another in the same non-obtuse triangle is ${\displaystyle 2{\sqrt {2}}/3=0.94....}$[26] Both of these extreme cases occur for the isosceles right triangle.

From an interior point in a reference triangle, the nearest points on the three sides serve as the vertices of the pedal triangle of that point. If the interior point is the circumcenter of the reference triangle, the vertices of the pedal triangle are the midpoints of the reference triangle's sides, and so the pedal triangle is called the midpoint triangle or medial triangle. The midpoint triangle subdivides the reference triangle into four congruent triangles which are similar to the reference triangle.

The Gergonne triangle or intouch triangle of a reference triangle has its vertices at the three points of tangency of the reference triangle's sides with its incircle. The extouch triangle of a reference triangle has its vertices at the points of tangency of the reference triangle's excircles with its sides (not extended).