# Triangle

For other uses, see Triangle (disambiguation). Triangle_sentence_1

Triangle_table_infobox_0

Symmetry groupTriangle_header_cell_0_5_0 Dihedral (D3), order 2×3Triangle_cell_0_5_1
PropertiesTriangle_header_cell_0_8_0 Convex, cyclic, equilateral, isogonal, isotoxalTriangle_cell_0_8_1

Triangle_table_infobox_1

see belowTriangle_cell_1_3_1

Internal angle (degrees)Triangle_header_cell_1_4_0 60° (for equilateral)Triangle_cell_1_4_1

In Euclidean geometry, any three points, when non-, determine a unique triangle and simultaneously, a unique plane (i.e. a two-dimensional Euclidean space). Triangle_sentence_2

In other words, there is only one plane that contains that triangle, and every triangle is contained in some plane. Triangle_sentence_3

If the entire geometry is only the Euclidean plane, there is only one plane and all triangles are contained in it; however, in higher-dimensional Euclidean spaces, this is no longer true. Triangle_sentence_4

## Types of triangle Triangle_section_0

### By lengths of sides Triangle_section_1

Triangles can be classified according to the lengths of their sides: Triangle_sentence_6

Triangle_unordered_list_0

• An equilateral triangle has three sides of the same length. An equilateral triangle is also a regular polygon with all angles measuring 60°.Triangle_item_0_0
• An isosceles triangle has two sides of equal length. 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. 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.Triangle_item_0_1
• A scalene triangle has all its sides of different lengths. Equivalently, it has all angles of different measure.Triangle_item_0_2

Hatch marks, also called tick marks, are used in diagrams of triangles and other geometric figures to identify sides of equal lengths. Triangle_sentence_7

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. Triangle_sentence_8

In a triangle, the pattern is usually no more than 3 ticks. Triangle_sentence_9

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. Triangle_sentence_10

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. Triangle_sentence_11

### By internal angles Triangle_section_2

Triangles can also be classified according to their internal angles, measured here in degrees. Triangle_sentence_12

Triangle_unordered_list_1

• A right triangle (or right-angled triangle, formerly called a rectangled 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 (singular: ) 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: a + b = c, 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 3 + 4 = 5. 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).Triangle_item_1_3
• Triangles that do not have an angle measuring 90° are called oblique triangles.Triangle_item_1_4
• A triangle with all interior angles measuring less than 90° is an acute triangle or acute-angled triangle. If c is the length of the longest side, then a + b > c, where a and b are the lengths of the other sides.Triangle_item_1_5
• A triangle with one interior angle measuring more than 90° is an obtuse triangle or obtuse-angled triangle. If c is the length of the longest side, then a + b < c, where a and b are the lengths of the other sides.Triangle_item_1_6
• A triangle with an interior angle of 180° (and vertices) is degenerate.Triangle_item_1_7
• A right degenerate triangle has collinear vertices, two of which are coincident.Triangle_item_1_8

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. Triangle_sentence_13

It follows that in a triangle where all angles have the same measure, all three sides have the same length, and therefore is equilateral. Triangle_sentence_14

## Basic facts Triangle_section_3

Triangles are assumed to be two-dimensional plane figures, unless the context provides otherwise (see Non-planar triangles, below). Triangle_sentence_15

In rigorous treatments, a triangle is therefore called a 2-simplex (see also Polytope). Triangle_sentence_16

Elementary facts about triangles were presented by Euclid, in books 1–4 of his Elements, written around 300 BC. Triangle_sentence_17

The sum of the measures of the interior angles of a triangle in Euclidean space is always 180 degrees. Triangle_sentence_18

This fact is equivalent to Euclid's parallel postulate. Triangle_sentence_19

This allows determination of the measure of the third angle of any triangle, given the measure of two angles. Triangle_sentence_20

An exterior angle of a triangle is an angle that is a linear pair (and hence supplementary) to an interior angle. Triangle_sentence_21

The measure of an exterior angle of a triangle is equal to the sum of the measures of the two interior angles that are not adjacent to it; this is the exterior angle theorem. Triangle_sentence_22

The sum of the measures of the three exterior angles (one for each vertex) of any triangle is 360 degrees. Triangle_sentence_23

### Similarity and congruence Triangle_section_4

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. Triangle_sentence_24

The corresponding sides of similar triangles have lengths that are in the same proportion, and this property is also sufficient to establish similarity. Triangle_sentence_25

Some basic theorems about similar triangles are: Triangle_sentence_26

Triangle_unordered_list_2

• 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.Triangle_item_2_9
• 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.)Triangle_item_2_10
• If and only if three pairs of corresponding sides of two triangles are all in the same proportion, then the triangles are similar.Triangle_item_2_11

Two triangles that are congruent have exactly the same size and shape: all pairs of corresponding interior angles are equal in measure, and all pairs of corresponding sides have the same length. Triangle_sentence_27

(This is a total of six equalities, but three are often sufficient to prove congruence.) Triangle_sentence_28

Some individually necessary and sufficient conditions for a pair of triangles to be congruent are: Triangle_sentence_29

Triangle_unordered_list_3

• 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.Triangle_item_3_12
• 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.)Triangle_item_3_13
• SSS: Each side of a triangle has the same length as a corresponding side of the other triangle.Triangle_item_3_14
• 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.)Triangle_item_3_15

Some individually sufficient conditions are: Triangle_sentence_30

Triangle_unordered_list_4

• 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).Triangle_item_4_16
• 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.Triangle_item_4_17

An important condition is: Triangle_sentence_31

Triangle_unordered_list_5

• 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.Triangle_item_5_18

Using right triangles and the concept of similarity, the trigonometric functions sine and cosine can be defined. Triangle_sentence_32

These are functions of an angle which are investigated in trigonometry. Triangle_sentence_33

### Right triangles Triangle_section_5

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. Triangle_sentence_34

If the hypotenuse has length c, and the legs have lengths a and b, then the theorem states that Triangle_sentence_35

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. Triangle_sentence_36

Some other facts about right triangles: Triangle_sentence_37

Triangle_unordered_list_6

• The acute angles of a right triangle are complementary.Triangle_item_6_19

Triangle_unordered_list_7

• 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.Triangle_item_7_20
• 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:Triangle_item_7_21

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). Triangle_sentence_38

## Existence of a triangle Triangle_section_6

### Condition on the sides Triangle_section_7

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. Triangle_sentence_39

That sum can equal the length of the third side only in the case of a degenerate triangle, one with collinear vertices. Triangle_sentence_40

It is not possible for that sum to be less than the length of the third side. Triangle_sentence_41

A triangle with three given positive side lengths exists if and only if those side lengths satisfy the triangle inequality. Triangle_sentence_42

### Conditions on the angles Triangle_section_8

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°. Triangle_sentence_43

If degenerate triangles are permitted, angles of 0° are permitted. Triangle_sentence_44

#### Trigonometric conditions Triangle_section_9

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: Triangle_sentence_45

the last equality applying only if none of the angles is 90° (so the tangent function's value is always finite). Triangle_sentence_46

## Points, lines, and circles associated with a triangle Triangle_section_10

There are thousands of different constructions that find a special point associated with (and often inside) a triangle, satisfying some unique property: see the article Encyclopedia of Triangle Centers for a catalogue of them. Triangle_sentence_47

Often they are constructed by finding three lines associated in a symmetrical way with the three sides (or vertices) and then proving that the three lines meet in a single point: an important tool for proving the existence of these is Ceva's theorem, which gives a criterion for determining when three such lines are concurrent. Triangle_sentence_48

Similarly, lines associated with a triangle are often constructed by proving that three symmetrically constructed points are : here Menelaus' theorem gives a useful general criterion. Triangle_sentence_49

In this section just a few of the most commonly encountered constructions are explained. Triangle_sentence_50

A perpendicular bisector of a side of a triangle is a straight line passing through the midpoint of the side and being perpendicular to it, i.e. forming a right angle with it. Triangle_sentence_51

The three perpendicular bisectors meet in a single point, the triangle's circumcenter, usually denoted by O; this point is the center of the circumcircle, the circle passing through all three vertices. Triangle_sentence_52

The diameter of this circle, called the circumdiameter, can be found from the law of sines stated above. Triangle_sentence_53

Thales' theorem implies that if the circumcenter is located on a side of the triangle, then the opposite angle is a right one. Triangle_sentence_55

If the circumcenter is located inside the triangle, then the triangle is acute; if the circumcenter is located outside the triangle, then the triangle is obtuse. Triangle_sentence_56

An altitude of a triangle is a straight line through a vertex and perpendicular to (i.e. forming a right angle with) the opposite side. Triangle_sentence_57

This opposite side is called the base of the altitude, and the point where the altitude intersects the base (or its extension) is called the foot of the altitude. Triangle_sentence_58

The length of the altitude is the distance between the base and the vertex. Triangle_sentence_59

The three altitudes intersect in a single point, called the orthocenter of the triangle, usually denoted by H. The orthocenter lies inside the triangle if and only if the triangle is acute. Triangle_sentence_60

An angle bisector of a triangle is a straight line through a vertex which cuts the corresponding angle in half. Triangle_sentence_61

The three angle bisectors intersect in a single point, the incenter, usually denoted by I, the center of the triangle's incircle. Triangle_sentence_62

The incircle is the circle which lies inside the triangle and touches all three sides. Triangle_sentence_63

There are three other important circles, the excircles; they lie outside the triangle and touch one side as well as the extensions of the other two. Triangle_sentence_65

The centers of the in- and excircles form an orthocentric system. Triangle_sentence_66

A median of a triangle is a straight line through a vertex and the midpoint of the opposite side, and divides the triangle into two equal areas. Triangle_sentence_67

The three medians intersect in a single point, the triangle's centroid or geometric barycenter, usually denoted by G. The centroid of a rigid triangular object (cut out of a thin sheet of uniform density) is also its center of mass: the object can be balanced on its centroid in a uniform gravitational field. Triangle_sentence_68

The centroid cuts every median in the ratio 2:1, i.e. the distance between a vertex and the centroid is twice the distance between the centroid and the midpoint of the opposite side. Triangle_sentence_69

The midpoints of the three sides and the feet of the three altitudes all lie on a single circle, the triangle's nine-point circle. Triangle_sentence_70

The remaining three points for which it is named are the midpoints of the portion of altitude between the vertices and the orthocenter. Triangle_sentence_71

The radius of the nine-point circle is half that of the circumcircle. Triangle_sentence_72

It touches the incircle (at the Feuerbach point) and the three excircles. Triangle_sentence_73

The orthocenter (blue point), center of the nine-point circle (red), centroid (orange), and circumcenter (green) all lie on a single line, known as Euler's line (red line). Triangle_sentence_74

The center of the nine-point circle lies at the midpoint between the orthocenter and the circumcenter, and the distance between the centroid and the circumcenter is half that between the centroid and the orthocenter. Triangle_sentence_75

The center of the incircle is not in general located on Euler's line. Triangle_sentence_76

If one reflects a median in the angle bisector that passes through the same vertex, one obtains a symmedian. Triangle_sentence_77

The three symmedians intersect in a single point, the symmedian point of the triangle. Triangle_sentence_78

## Computing the sides and angles Triangle_section_11

There are various standard methods for calculating the length of a side or the measure of an angle. Triangle_sentence_79

Certain methods are suited to calculating values in a right-angled triangle; more complex methods may be required in other situations. Triangle_sentence_80

### Trigonometric ratios in right triangles Triangle_section_12

Main article: Trigonometric functions Triangle_sentence_81

In right triangles, the trigonometric ratios of sine, cosine and tangent can be used to find unknown angles and the lengths of unknown sides. Triangle_sentence_82

The sides of the triangle are known as follows: Triangle_sentence_83

Triangle_unordered_list_8

• The hypotenuse is the side opposite the right angle, or defined as the longest side of a right-angled triangle, in this case h.Triangle_item_8_22
• The opposite side is the side opposite to the angle we are interested in, in this case a.Triangle_item_8_23
• 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.Triangle_item_8_24

#### Sine, cosine and tangent Triangle_section_13

The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. Triangle_sentence_84

In our case Triangle_sentence_85

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. Triangle_sentence_86

The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse. Triangle_sentence_87

In our case Triangle_sentence_88

The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side. Triangle_sentence_89

In our case Triangle_sentence_90

The acronym "SOH-CAH-TOA" is a useful mnemonic for these ratios. Triangle_sentence_91

#### Inverse functions Triangle_section_14

The inverse trigonometric functions can be used to calculate the internal angles for a right angled triangle with the length of any two sides. Triangle_sentence_92

Arcsin can be used to calculate an angle from the length of the opposite side and the length of the hypotenuse. Triangle_sentence_93

Arccos can be used to calculate an angle from the length of the adjacent side and the length of the hypotenuse. Triangle_sentence_94

Arctan can be used to calculate an angle from the length of the opposite side and the length of the adjacent side. Triangle_sentence_95

In introductory geometry and trigonometry courses, the notation sin, cos, etc., are often used in place of arcsin, arccos, etc. Triangle_sentence_96

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. Triangle_sentence_97

### Sine, cosine and tangent rules Triangle_section_15

Main articles: Law of sines, Law of cosines, and Law of tangents Triangle_sentence_98

The law of sines, or sine rule, states that the ratio of the length of a side to the sine of its corresponding opposite angle is constant, that is Triangle_sentence_99

This ratio is equal to the diameter of the circumscribed circle of the given triangle. Triangle_sentence_100

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. Triangle_sentence_101

This triangle can be constructed by first constructing a circle of diameter 1, and inscribing in it two of the angles of the triangle. Triangle_sentence_102

The length of the sides of that triangle will be sin α, sin β and sin γ. Triangle_sentence_103

The side whose length is sin α is opposite to the angle whose measure is α, etc. Triangle_sentence_104

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. Triangle_sentence_105

As per the law: Triangle_sentence_106

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: Triangle_sentence_107

If the lengths of all three sides of any triangle are known the three angles can be calculated: Triangle_sentence_108

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. Triangle_sentence_109

It states that: Triangle_sentence_110

### Solution of triangles Triangle_section_16

Main article: Solution of triangles Triangle_sentence_111

"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. Triangle_sentence_112

The triangle can be located on a plane or on a sphere. Triangle_sentence_113

This problem often occurs in various trigonometric applications, such as geodesy, astronomy, construction, navigation etc. Triangle_sentence_114

## Computing the area of a triangle Triangle_section_17

Calculating the area T of a triangle is an elementary problem encountered often in many different situations. Triangle_sentence_116

The best known and simplest formula is: Triangle_sentence_117

where b is the length of the base of the triangle, and h is the height or altitude of the triangle. Triangle_sentence_118

The term "base" denotes any side, and "height" denotes the length of a perpendicular from the vertex opposite the base onto the line containing the base. Triangle_sentence_119

In 499 CE Aryabhata, used this illustrated method in the Aryabhatiya (section 2.6). Triangle_sentence_120

Although simple, this formula is only useful if the height can be readily found, which is not always the case. Triangle_sentence_121

For example, the surveyor of a triangular field might find it relatively easy to measure the length of each side, but relatively difficult to construct a 'height'. Triangle_sentence_122

Various methods may be used in practice, depending on what is known about the triangle. Triangle_sentence_123

The following is a selection of frequently used formulae for the area of a triangle. Triangle_sentence_124

### Using trigonometry Triangle_section_18

The height of a triangle can be found through the application of trigonometry. Triangle_sentence_125

Knowing AAS: Triangle_sentence_126

and analogously if the known side is a or c. Triangle_sentence_127

Knowing ASA: Triangle_sentence_128

and analogously if the known side is b or c. Triangle_sentence_129

### Using Heron's formula Triangle_section_19

The shape of the triangle is determined by the lengths of the sides. Triangle_sentence_130

Therefore, the area can also be derived from the lengths of the sides. Triangle_sentence_131

By Heron's formula: Triangle_sentence_132

Three other equivalent ways of writing Heron's formula are Triangle_sentence_133

### Using vectors Triangle_section_20

The area of a parallelogram embedded in a three-dimensional Euclidean space can be calculated using vectors. Triangle_sentence_134

Let vectors AB and AC point respectively from A to B and from A to C. The area of parallelogram ABDC is then Triangle_sentence_135

which is the magnitude of the cross product of vectors AB and AC. Triangle_sentence_136

The area of triangle ABC is half of this, Triangle_sentence_137

The area of triangle ABC can also be expressed in terms of dot products as follows: Triangle_sentence_138

In two-dimensional Euclidean space, expressing vector AB as a free vector in Cartesian space equal to (x1,y1) and AC as (x2,y2), this can be rewritten as: Triangle_sentence_139

### Using coordinates Triangle_section_21

If vertex A is located at the origin (0, 0) of a Cartesian coordinate system and the coordinates of the other two vertices are given by B = (xB, yB) and C = (xC, yC), then the area can be computed as ​⁄2 times the absolute value of the determinant Triangle_sentence_140

For three general vertices, the equation is: Triangle_sentence_141

which can be written as Triangle_sentence_142

If the points are labeled sequentially in the counterclockwise direction, the above determinant expressions are positive and the absolute value signs can be omitted. Triangle_sentence_143

The above formula is known as the shoelace formula or the surveyor's formula. Triangle_sentence_144

is equivalent to the shoelace formula. Triangle_sentence_145

In three dimensions, the area of a general triangle A = (xA, yA, zA), B = (xB, yB, zB) and C = (xC, yC, zC) is the Pythagorean sum of the areas of the respective projections on the three principal planes (i.e. x = 0, y = 0 and z = 0): Triangle_sentence_146

### Using line integrals Triangle_section_22

The area within any closed curve, such as a triangle, is given by the line integral around the curve of the algebraic or signed distance of a point on the curve from an arbitrary oriented straight line L. Points to the right of L as oriented are taken to be at negative distance from L, while the weight for the integral is taken to be the component of arc length parallel to L rather than arc length itself. Triangle_sentence_147

This method is well suited to computation of the area of an arbitrary polygon. Triangle_sentence_148

Taking L to be the x-axis, the line integral between consecutive vertices (xi,yi) and (xi+1,yi+1) is given by the base times the mean height, namely (xi+1 − xi)(yi + yi+1)/2. Triangle_sentence_149

The sign of the area is an overall indicator of the direction of traversal, with negative area indicating counterclockwise traversal. Triangle_sentence_150

The area of a triangle then falls out as the case of a polygon with three sides. Triangle_sentence_151

While the line integral method has in common with other coordinate-based methods the arbitrary choice of a coordinate system, unlike the others it makes no arbitrary choice of vertex of the triangle as origin or of side as base. Triangle_sentence_152

Furthermore, the choice of coordinate system defined by L commits to only two degrees of freedom rather than the usual three, since the weight is a local distance (e.g. xi+1 − xi in the above) whence the method does not require choosing an axis normal to L. Triangle_sentence_153

When working in polar coordinates it is not necessary to convert to Cartesian coordinates to use line integration, since the line integral between consecutive vertices (ri,θi) and (ri+1,θi+1) of a polygon is given directly by riri+1sin(θi+1 − θi)/2. Triangle_sentence_154

This is valid for all values of θ, with some decrease in numerical accuracy when |θ| is many orders of magnitude greater than π. Triangle_sentence_155

With this formulation negative area indicates clockwise traversal, which should be kept in mind when mixing polar and cartesian coordinates. Triangle_sentence_156

Just as the choice of y-axis (x = 0) is immaterial for line integration in cartesian coordinates, so is the choice of zero heading (θ = 0) immaterial here. Triangle_sentence_157

### Formulas resembling Heron's formula Triangle_section_23

Three formulas have the same structure as Heron's formula but are expressed in terms of different variables. Triangle_sentence_158

First, denoting the medians from sides a, b, and c respectively as ma, mb, and mc and their semi-sum (ma + mb + mc)/2 as σ, we have Triangle_sentence_159

And denoting the semi-sum of the angles' sines as S = [(sin α) + (sin β) + (sin γ)]/2, we have Triangle_sentence_160

### Using Pick's theorem Triangle_section_24

See Pick's theorem for a technique for finding the area of any arbitrary lattice polygon (one drawn on a grid with vertically and horizontally adjacent lattice points at equal distances, and with vertices on lattice points). Triangle_sentence_161

The theorem states: Triangle_sentence_162

### Other area formulas Triangle_section_25

Numerous other area formulas exist, such as Triangle_sentence_163

where r is the inradius, and s is the semiperimeter (in fact, this formula holds for all tangential polygons), and Triangle_sentence_164

We also have Triangle_sentence_165

and Triangle_sentence_166

for circumdiameter D; and Triangle_sentence_167

for angle α ≠ 90°. Triangle_sentence_168

The area can also be expressed as Triangle_sentence_169

In 1885, Baker gave a collection of over a hundred distinct area formulas for the triangle. Triangle_sentence_170

These include: Triangle_sentence_171

### Upper bound on the area Triangle_section_26

The area T of any triangle with perimeter p satisfies Triangle_sentence_173

with equality holding if and only if the triangle is equilateral. Triangle_sentence_174

Other upper bounds on the area T are given by Triangle_sentence_175

and Triangle_sentence_176

both again holding if and only if the triangle is equilateral. Triangle_sentence_177

### Bisecting the area Triangle_section_27

There are infinitely many lines that bisect the area of a triangle. Triangle_sentence_178

Three of them are the medians, which are the only area bisectors that go through the centroid. Triangle_sentence_179

Three other area bisectors are parallel to the triangle's sides. Triangle_sentence_180

Any line through a triangle that splits both the triangle's area and its perimeter in half goes through the triangle's incenter. Triangle_sentence_181

There can be one, two, or three of these for any given triangle. Triangle_sentence_182

## Further formulas for general Euclidean triangles Triangle_section_28

The formulas in this section are true for all Euclidean triangles. Triangle_sentence_184

### Medians, angle bisectors, perpendicular side bisectors, and altitudes Triangle_section_29

Main articles: Median (geometry), Angle bisector, Bisection § Perpendicular bisectors, and Altitude (geometry) Triangle_sentence_185

The medians and the sides are related by Triangle_sentence_186

and Triangle_sentence_187

and equivalently for mb and mc. Triangle_sentence_188

For angle A opposite side a, the length of the internal angle bisector is given by Triangle_sentence_189

for semiperimeter s, where the bisector length is measured from the vertex to where it meets the opposite side. Triangle_sentence_190

The interior perpendicular bisectors are given by Triangle_sentence_191

The altitude from, for example, the side of length a is Triangle_sentence_192

The following formulas involve the circumradius R and the inradius r: Triangle_sentence_194

where ha etc. are the altitudes to the subscripted sides; Triangle_sentence_195

and Triangle_sentence_196

The product of two sides of a triangle equals the altitude to the third side times the diameter D of the circumcircle: Triangle_sentence_197

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). Triangle_sentence_198

Then Triangle_sentence_199

### Centroid Triangle_section_32

Main article: Centroid Triangle_sentence_200

Let G be the centroid of a triangle with vertices A, B, and C, and let P be any interior point. Triangle_sentence_201

Then the distances between the points are related by Triangle_sentence_202

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: Triangle_sentence_203

Let qa, qb, and qc be the distances from the centroid to the sides of lengths a, b, and c. Then Triangle_sentence_204

and Triangle_sentence_205

for area T. Triangle_sentence_206

### Circumcenter, incenter, and orthocenter Triangle_section_33

Main articles: Circumcenter, Incenter, and Orthocenter Triangle_sentence_207

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. Triangle_sentence_208

Here a segment's length is considered to be negative if and only if the segment lies entirely outside the triangle. Triangle_sentence_209

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. Triangle_sentence_210

Euler's theorem states that the distance d between the circumcenter and the incenter is given by Triangle_sentence_211

or equivalently Triangle_sentence_212

Thus for all triangles R ≥ 2r, with equality holding for equilateral triangles. Triangle_sentence_214

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. Triangle_sentence_215

The distance from a side to the circumcenter equals half the distance from the opposite vertex to the orthocenter. Triangle_sentence_216

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: Triangle_sentence_217

### Angles Triangle_section_34

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 Triangle_sentence_218

### Morley's trisector theorem Triangle_section_35

Main article: Morley's trisector theorem Triangle_sentence_219

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. Triangle_sentence_220

## Figures inscribed in a triangle Triangle_section_36

### Conics Triangle_section_37

As discussed above, every triangle has a unique inscribed circle (incircle) that is interior to the triangle and tangent to all three sides. Triangle_sentence_221

Every triangle has a unique Steiner inellipse which is interior to the triangle and tangent at the midpoints of the sides. Triangle_sentence_222

Marden's theorem shows how to find the foci of this ellipse. Triangle_sentence_223

This ellipse has the greatest area of any ellipse tangent to all three sides of the triangle. Triangle_sentence_224

The Mandart inellipse of a triangle is the ellipse inscribed within the triangle tangent to its sides at the contact points of its excircles. Triangle_sentence_225

For any ellipse inscribed in a triangle ABC, let the foci be P and Q. Triangle_sentence_226

Then Triangle_sentence_227

### Convex polygon Triangle_section_38

Every convex polygon with area T can be inscribed in a triangle of area at most equal to 2T. Triangle_sentence_228

Equality holds (exclusively) for a parallelogram. Triangle_sentence_229

### Hexagon Triangle_section_39

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. Triangle_sentence_230

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. Triangle_sentence_231

### Squares Triangle_section_40

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). Triangle_sentence_232

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. Triangle_sentence_233

An obtuse triangle has only one inscribed square, with a side coinciding with part of the triangle's longest side. Triangle_sentence_234

Within a given triangle, a longer common side is associated with a smaller inscribed square. Triangle_sentence_235

If an inscribed square has side of length qa and the triangle has a side of length a, part of which side coincides with a side of the square, then qa, a, the altitude ha from the side a, and the triangle's area T are related according to Triangle_sentence_236

### Triangles Triangle_section_41

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. Triangle_sentence_237

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. Triangle_sentence_238

The midpoint triangle subdivides the reference triangle into four congruent triangles which are similar to the reference triangle. Triangle_sentence_239

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. Triangle_sentence_240

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). Triangle_sentence_241

## Figures circumscribed about a triangle Triangle_section_42

The tangential triangle of a reference triangle (other than a right triangle) is the triangle whose sides are on the tangent lines to the reference triangle's circumcircle at its vertices. Triangle_sentence_242

As mentioned above, every triangle has a unique circumcircle, a circle passing through all three vertices, whose center is the intersection of the perpendicular bisectors of the triangle's sides. Triangle_sentence_243

Further, every triangle has a unique Steiner circumellipse, which passes through the triangle's vertices and has its center at the triangle's centroid. Triangle_sentence_244

Of all ellipses going through the triangle's vertices, it has the smallest area. Triangle_sentence_245

The Kiepert hyperbola is the unique conic which passes through the triangle's three vertices, its centroid, and its circumcenter. Triangle_sentence_246

Of all triangles contained in a given convex polygon, there exists a triangle with maximal area whose vertices are all vertices of the given polygon. Triangle_sentence_247

## Specifying the location of a point in a triangle Triangle_section_43

One way to identify locations of points in (or outside) a triangle is to place the triangle in an arbitrary location and orientation in the Cartesian plane, and to use Cartesian coordinates. Triangle_sentence_248

While convenient for many purposes, this approach has the disadvantage of all points' coordinate values being dependent on the arbitrary placement in the plane. Triangle_sentence_249

Two systems avoid that feature, so that the coordinates of a point are not affected by moving the triangle, rotating it, or reflecting it as in a mirror, any of which give a congruent triangle, or even by rescaling it to give a similar triangle: Triangle_sentence_250

## Non-planar triangles Triangle_section_44

A non-planar triangle is a triangle which is not contained in a (flat) plane. Triangle_sentence_251

Some examples of non-planar triangles in non-Euclidean geometries are spherical triangles in spherical geometry and hyperbolic triangles in hyperbolic geometry. Triangle_sentence_252

While the measures of the internal angles in planar triangles always sum to 180°, a hyperbolic triangle has measures of angles that sum to less than 180°, and a spherical triangle has measures of angles that sum to more than 180°. Triangle_sentence_253

A hyperbolic triangle can be obtained by drawing on a negatively curved surface, such as a saddle surface, and a spherical triangle can be obtained by drawing on a positively curved surface such as a sphere. Triangle_sentence_254

Thus, if one draws a giant triangle on the surface of the Earth, one will find that the sum of the measures of its angles is greater than 180°; in fact it will be between 180° and 540°. Triangle_sentence_255

In particular it is possible to draw a triangle on a sphere such that the measure of each of its internal angles is equal to 90°, adding up to a total of 270°. Triangle_sentence_256

Specifically, on a sphere the sum of the angles of a triangle is Triangle_sentence_257

Triangle_description_list_9

• 180° × (1 + 4f),Triangle_item_9_25

where f is the fraction of the sphere's area which is enclosed by the triangle. Triangle_sentence_258

For example, suppose that we draw a triangle on the Earth's surface with vertices at the North Pole, at a point on the equator at 0° longitude, and a point on the equator at 90° West longitude. Triangle_sentence_259

The great circle line between the latter two points is the equator, and the great circle line between either of those points and the North Pole is a line of longitude; so there are right angles at the two points on the equator. Triangle_sentence_260

Moreover, the angle at the North Pole is also 90° because the other two vertices differ by 90° of longitude. Triangle_sentence_261

So the sum of the angles in this triangle is 90° + 90° + 90° = 270°. Triangle_sentence_262

The triangle encloses 1/4 of the northern hemisphere (90°/360° as viewed from the North Pole) and therefore 1/8 of the Earth's surface, so in the formula f = 1/8; thus the formula correctly gives the sum of the triangle's angles as 270°. Triangle_sentence_263

From the above angle sum formula we can also see that the Earth's surface is locally flat: If we draw an arbitrarily small triangle in the neighborhood of one point on the Earth's surface, the fraction f of the Earth's surface which is enclosed by the triangle will be arbitrarily close to zero. Triangle_sentence_264

In this case the angle sum formula simplifies to 180°, which we know is what Euclidean geometry tells us for triangles on a flat surface. Triangle_sentence_265

## Triangles in construction Triangle_section_45

Main article: Truss Triangle_sentence_266

Rectangles have been the most popular and common geometric form for buildings since the shape is easy to stack and organize; as a standard, it is easy to design furniture and fixtures to fit inside rectangularly shaped buildings. Triangle_sentence_267

But triangles, while more difficult to use conceptually, provide a great deal of strength. Triangle_sentence_268

As computer technology helps architects design creative new buildings, triangular shapes are becoming increasingly prevalent as parts of buildings and as the primary shape for some types of skyscrapers as well as building materials. Triangle_sentence_269

In Tokyo in 1989, architects had wondered whether it was possible to build a 500-story tower to provide affordable office space for this densely packed city, but with the danger to buildings from earthquakes, architects considered that a triangular shape would be necessary if such a building were to be built. Triangle_sentence_270

In New York City, as Broadway crisscrosses major avenues, the resulting blocks are cut like triangles, and buildings have been built on these shapes; one such building is the triangularly shaped Flatiron Building which real estate people admit has a "warren of awkward spaces that do not easily accommodate modern office furniture" but that has not prevented the structure from becoming a landmark icon. Triangle_sentence_271

Designers have made houses in Norway using triangular themes. Triangle_sentence_272

Triangle shapes have appeared in churches as well as public buildings including colleges as well as supports for innovative home designs. Triangle_sentence_273

Triangles are sturdy; while a rectangle can collapse into a parallelogram from pressure to one of its points, triangles have a natural strength which supports structures against lateral pressures. Triangle_sentence_274

A triangle will not change shape unless its sides are bent or extended or broken or if its joints break; in essence, each of the three sides supports the other two. Triangle_sentence_275

A rectangle, in contrast, is more dependent on the strength of its joints in a structural sense. Triangle_sentence_276

Some innovative designers have proposed making bricks not out of rectangles, but with triangular shapes which can be combined in three dimensions. Triangle_sentence_277

It is likely that triangles will be used increasingly in new ways as architecture increases in complexity. Triangle_sentence_278

It is important to remember that triangles are strong in terms of rigidity, but while packed in a tessellating arrangement triangles are not as strong as hexagons under compression (hence the prevalence of hexagonal forms in nature). Triangle_sentence_279

Tessellated triangles still maintain superior strength for cantilevering however, and this is the basis for one of the strongest man made structures, the tetrahedral truss. Triangle_sentence_280