Editing Triangle (section)

== Properties ==
=== Points, lines, and circles associated with a triangle ===
{{main article|Encyclopedia of Triangle Centers}}
Each triangle has many special points inside it, on its edges, or otherwise associated with it. They are constructed by finding three lines associated symmetrically 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 points is [[Ceva's theorem]], which gives a criterion for determining when three such lines are [[concurrent lines|concurrent]].{{sfn|Holme|2010|p=[https://books.google.com/books?id=zXwQGo8jyHUC&pg=PA210 210]}} Similarly, lines associated with a triangle are often constructed by proving that three symmetrically constructed points are [[collinear]]; here [[Menelaus' theorem]] gives a useful general criterion.{{sfn|Holme|2010|p=[https://books.google.com/books?id=zXwQGo8jyHUC&pg=PA143 143]}} In this section, just a few of the most commonly encountered constructions are explained.

A [[bisection|perpendicular bisector]] of a side of a triangle is a straight line passing through the [[midpoint]] of the side and being perpendicular to it, forming a right angle with it.{{sfn|Lang|Murrow|1988|p=[https://books.google.com/books?id=pc_kBwAAQBAJ&pg=PA126 126&ndash;127]}} The three perpendicular bisectors meet in a single point, the triangle's [[circumcenter]]; this point is the center of the [[circumcircle]], the circle passing through all three vertices.{{sfn|Lang|Murrow|1988|p=[https://books.google.com/books?id=pc_kBwAAQBAJ&pg=PA128 128]}} [[Thales' theorem]] implies that if the circumcenter is located on the side of the triangle, then the angle opposite that side is a right angle.{{sfn|Anglin|Lambek|1995|p=[https://books.google.com/books?id=flblBwAAQBAJ&pg=PA30 30]}} 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.{{sfn|Ryan|2008|p=[https://books.google.com/books?id=b_qM4HImlPgC&pg=PA105 105]}}
{{multipleimage
| align             = center
| total_width       = 600
| footer            = 
| image1            = Triangle.Circumcenter.svg
| image2            = Triangle.Incircle.svg
| image3            = Triangle.Centroid.svg
| image4            = Triangle.Orthocenter.svg
| caption1          = The intersection of perpendicular bisectors is the [[circumcenter]].
| caption2          = The intersection of the angle bisectors is the [[incenter]]
| caption3          = The intersection of the medians known as the [[centroid]]
| caption4          = The intersection of the altitudes is the [[orthocenter]]
}}

An [[altitude (triangle)|altitude]] of a triangle is a straight line through a vertex and perpendicular to the opposite side. 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.<ref>{{multiref
|{{harvnb|Lang|Murrow|1988|p=[https://books.google.com/books?id=pc_kBwAAQBAJ&pg=PA84 84]}}
|{{harvnb|King|2021|p=[https://books.google.com/books?id=6UgrEAAAQBAJ&pg=PA78 78]}}
}}</ref> The length of the altitude is the distance between the base and the vertex. The three altitudes intersect in a single point, called the [[orthocenter]] of the triangle.{{sfn|King|2021|p=[https://books.google.com/books?id=6UgrEAAAQBAJ&pg=PA153 153]}} The orthocenter lies inside the triangle if and only if the triangle is acute.{{sfn|Ryan|2008|p=[https://books.google.com/books?id=b_qM4HImlPgC&pg=PA106 106]}}

{{multiple image
 | total_width = 400
 | image1 = Triangle.NinePointCircle.svg
 | image2 = Triangle.EulerLine.svg
 | footer = [[Nine-point circle]] demonstrates a symmetry where six points lie on the edge of the triangle. [[Euler's line]] is a straight line through the orthocenter (blue), the center of the nine-point circle (red), centroid (orange), and circumcenter (green).
}}
An [[angle bisector]] of a triangle is a straight line through a vertex that cuts the corresponding angle in half. The three angle bisectors intersect in a single point, the [[incenter]], which is the center of the triangle's [[incircle]]. The incircle is the circle that lies inside the triangle and touches all three sides. Its radius is called the inradius. There are three other important circles, the [[excircle]]s; they lie outside the triangle and touch one side, as well as the extensions of the other two. The centers of the incircles and excircles form an [[orthocentric system]].{{sfn|Ryan|2008|p=[https://books.google.com/books?id=b_qM4HImlPgC&pg=PA104 104]}} 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]].{{sfn|King|2021|p=[https://books.google.com/books?id=6UgrEAAAQBAJ&pg=PA155 155]}} The remaining three points for which it is named are the midpoints of the portion of altitude between the vertices and the [[orthocenter]]. The radius of the nine-point circle is half that of the circumcircle. It touches the incircle (at the [[Nine-point circle|Feuerbach point]]) and the three [[excircle]]s. The orthocenter (blue point), the center of the nine-point circle (red), the centroid (orange), and the circumcenter (green) all lie on a single line, known as [[Euler's line]] (red line). 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.{{sfn|King|2021|p=[https://books.google.com/books?id=6UgrEAAAQBAJ&pg=PA155 155]}} Generally, the incircle's center is not located on Euler's line.<ref>{{cite book | url=https://books.google.com/books?id=lR0SDnl2bPwC&pg=PA4 | title=Geometry Turned On: Dynamic Software in Learning, Teaching, and Research | publisher=The Mathematical Association of America |author1=Schattschneider, Doris |author2=King, James | year=1997 | pages=3–4 | isbn=978-0883850992}}</ref><ref>{{cite journal | last1 = Edmonds | first1 = Allan L. | last2 = Hajja | first2 = Mowaffaq | last3 = Martini | first3 = Horst | doi = 10.1007/s00025-008-0294-4 | issue = 1–2 | journal = [[Results in Mathematics]] | mr = 2430410 | pages = 41–50 | quote = It is well known that the incenter of a Euclidean triangle lies on its Euler line connecting the centroid and the circumcenter if and only if the triangle is isosceles. | title = Orthocentric simplices and biregularity | volume = 52 | year = 2008 }}</ref>

A [[median (geometry)|median]] of a triangle is a straight line through a [[vertex (geometry)|vertex]] and the [[midpoint]] of the opposite side, and divides the triangle into two equal areas. The three medians intersect in a single point, the triangle's [[centroid]] or geometric barycenter. 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.{{sfn|Ryan|2008|p=[https://books.google.com/books?id=b_qM4HImlPgC&pg=PA102 102]}} 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. If one reflects a median in the angle bisector that passes through the same vertex, one obtains a ''[[symmedian]]''. The three symmedians intersect in a single point, the [[symmedian point]] of the triangle.{{sfn|Holme|2010|p=[https://books.google.com/books?id=zXwQGo8jyHUC&pg=PA240 240]}}

=== Angles ===
[[File:Triangle sommeangles.svg|thumb|The measures of the interior angles of the triangle always add up to 180 degrees (same color to point out they are equal).]]
The [[Sum of angles of a triangle|sum of the measures of the interior angles of a triangle]] in [[Euclidean space]] is always 180 degrees.{{sfn|Heath|1926|loc=Proposition 32}} This fact is equivalent to Euclid's [[parallel postulate]]. This allows the determination of the measure of the third angle of any triangle, given the measure of two angles.{{sfn|Gonick|2024|pages=107–109}} An ''[[exterior angle]]'' of a triangle is an angle that is a linear pair (and hence [[supplementary angle|supplementary]]) to an interior angle. 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]].{{sfn|Ramsay|Richtmyer|1995|p=[https://books.google.com/books?id=4CDpBwAAQBAJ&pg=PA38 38]}} The sum of the measures of the three exterior angles (one for each vertex) of any triangle is 360 degrees, and indeed, this is true for any convex polygon, no matter how many sides it has.{{sfn|Gonick|2024|pages=224–225}}

Another relation between the internal angles and triangles creates a new concept of [[trigonometric function]]s. The primary trigonometric functions are [[sine and cosine]], as well as the other functions. They can be defined as the [[Sine and cosine#Right-angled triangle definition|ratio between any two sides of a right triangle]].{{sfn|Young|2017|p=[https://books.google.com/books?id=476ZDwAAQBAJ&pg=PA27 27]}} In a scalene triangle, the trigonometric functions can be used to find the unknown measure of either a side or an internal angle; methods for doing so use the [[law of sines]] and the [[law of cosines]].{{sfn|Axler|2012|p=[https://books.google.com/books?id=B5RxDwAAQBAJ&pg=PA634 634]}}

Any three angles that add to 180° can be the internal angles of a triangle. Infinitely many triangles have the same angles, since specifying the angles of a triangle does not determine its size. (A [[Degeneracy (mathematics)#Triangle|degenerate triangle]], whose vertices are [[collinearity|collinear]], has internal angles of 0° and 180°; whether such a shape counts as a triangle is a matter of convention.<ref>{{cite journal
 | last1 = Richmond | first1 = Bettina | author1-link = Bettina Richmond
 | last2 = Richmond | first2 = Thomas
 | doi = 10.1080/00029890.1997.11990706
 | issue = 8
 | journal = [[The American Mathematical Monthly]]
 | jstor = 2975234
 | mr = 1476755
 | pages = 713–719
 | title = Metric spaces in which all triangles are degenerate
 | volume = 104
 | year = 1997}}</ref><ref>{{cite thesis|type=PhD thesis|title=Making sense of definitions in geometry: Metric-combinatorial approaches to classifying triangles and quadrilaterals|last=Alonso|first=Orlando Braulio|publisher=Teachers College, Columbia University|year=2009|id={{ProQuest|304870039}}|page=57}}</ref>) The conditions for three angles <math> \alpha </math>, <math> \beta </math>, and <math> \gamma </math>, each of them between 0° and 180°, to be the angles of a triangle can also be stated using trigonometric functions. For example, a triangle with angles <math> \alpha </math>, <math> \beta </math>, and <math> \gamma </math> exists [[if and only if]]<ref>{{multiref
|{{harvnb|Verdiyan|Salas|2007}}
|{{harvnb|Longuet-Higgins|2003}}
}}</ref>
<math display="block"> \cos^2\alpha+\cos^2\beta+\cos^2\gamma+2\cos(\alpha)\cos(\beta)\cos(\gamma) = 1.</math>

=== Similarity and congruence ===
[[File:Angle-angle-side_triangle_congruence.svg|thumb|This diagram illustrates the geometric principle of angle-angle-side triangle congruence: given triangle ABC and triangle A'B'C', triangle ABC is congruent with triangle A'B'C' if and only if: angle CAB is congruent with angle C'A'B', and angle ABC is congruent with angle A'B'C', and BC is congruent with B'C'. Note [[Hatch_mark#Congruency_notation|hatch marks]] are used here to show angle and side equalities.]]
Two triangles are said to be ''[[similarity (geometry)|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.{{sfn|Gonick|2024|pages=157–167}}

Some basic [[theorem]]s 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.{{sfn|Gonick|2024|page=167}}
* If and only if one pair of corresponding sides of two triangles are in the same proportion as another pair of corresponding sides, and their included angles have the same measure, then the triangles are similar.{{sfn|Gonick|2024|page=171}} (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.{{efn|1=Again, in all cases "mirror images" are also similar.}}

Two triangles that are [[Congruence (geometry)|congruent]] have exactly the same size and shape. All pairs of congruent triangles are also similar, but not all pairs of similar triangles are congruent. Given two congruent triangles, 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.{{sfn|Gonick|2024|page=64}}

Some individually [[necessary and sufficient condition]]s for a pair of triangles to be congruent are:{{sfn|Gonick|2024|pages=65,72–73,111}}
* 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 side between them in a triangle have the same measure and length, respectively, as those in the other triangle. (This is the basis of [[Triangulation (surveying)|surveying by triangulation]].)
* SSS: Each side of a triangle has the same length as the 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.)

=== Area ===
{{main article|Area of a triangle}}
[[File:Triangle.GeometryArea.svg|upright=1.55|thumb|The area formula for a triangle can be proven by cutting two copies of the triangle into pieces and rearranging them into a rectangle.]]
In the Euclidean plane, [[area]] is defined by comparison with a square of side length {{tmath|1}}, which has area 1. There are several ways to calculate the area of an arbitrary triangle. One of the oldest and simplest is to take half the product of the length of one side {{tmath|b}} (the base) times the corresponding altitude {{tmath|h}}:{{sfn|Ryan|2008|p=[https://books.google.com/books?id=b_qM4HImlPgC&pg=PA98 98]}}
<math display="block"> T = \tfrac{1}{2}bh. </math>

This formula can be proven by cutting up the triangle and an identical copy into pieces and rearranging the pieces into the shape of a rectangle of base {{tmath|b}} and height {{tmath|h}}.

[[File:Triangle.TrigArea.svg|thumb|right|upright=0.8|Applying trigonometry to find the altitude {{math|1=''h''}}]]
If two sides {{tmath|a}} and {{tmath|b}} and their included angle <math> \gamma </math> are known, then the altitude can be calculated using trigonometry, {{tmath|1= h = a \sin(\gamma)}}, so the area of the triangle is:
<math display="block"> T = \tfrac{1}{2}ab \sin \gamma. </math>

[[Heron's formula]], named after [[Heron of Alexandria]], is a formula for finding the area of a triangle from the lengths of its sides <math> a </math>, <math> b </math>, <math> c </math>. Letting <math> s = \tfrac12(a + b + c) </math> be the  [[semiperimeter]],<ref>{{MacTutor|id=Heron |title=Heron of Alexandria}}</ref>
<math display="block"> T = \sqrt{s(s - a)(s - b)(s - c)}. </math>

[[File:Lexell's theorem in the plane.png|thumb|Orange triangles {{math|△''ABC''}} share a base {{mvar|AB}} and area. The locus of their apex {{mvar|C}} is a line (dashed green) parallel to the base. This is the Euclidean version of [[Lexell's theorem]].]]
Because the ratios between areas of shapes in the same plane are preserved by [[affine transformation]]s, the relative areas of triangles in any [[affine plane]] can be defined without reference to a notion of distance or squares. In any affine space (including Euclidean planes), every triangle with the same base and [[signed area|oriented area]] has its apex (the third vertex) on a line parallel to the base, and their common area is half of that of a [[parallelogram]] with the same base whose opposite side lies on the parallel line. This affine approach was developed in Book 1 of Euclid's ''Elements''.{{sfn|Heath|1926|loc=Propositions 36–41}}

Given [[affine coordinates]] (such as [[Cartesian coordinates]]) {{tmath|(x_A, y_A)}}, {{tmath|(x_B, y_B)}}, {{tmath|(x_C, y_C)}} for the vertices of a triangle, its relative oriented area can be calculated using the [[shoelace formula]],

<math display=block>\begin{align}
T &= \tfrac12 \begin{vmatrix}x_A & x_B & x_C \\ y_A & y_B & y_C \\ 1 & 1 & 1\end{vmatrix}
= \tfrac12 \begin{vmatrix} x_A & x_B \\ y_A & y_B \end{vmatrix}
+ \tfrac12 \begin{vmatrix} x_B & x_C \\ y_B & y_C \end{vmatrix}
+ \tfrac12 \begin{vmatrix} x_C & x_A \\ y_C & y_A \end{vmatrix} \\
&= \tfrac12(x_Ay_B - x_By_A + x_By_C - x_Cy_B + x_Cy_A - x_Ay_C),
\end{align}</math>

where <math>| \cdot |</math> is the [[matrix determinant]].<ref>{{cite journal |first=Bart |last=Braden |title=The Surveyor's Area Formula |journal=The College Mathematics Journal |volume=17 |issue=4 |year=1986 |pages=326–337 |url=https://www.maa.org/sites/default/files/pdf/pubs/Calc_Articles/ma063.pdf |doi=10.2307/2686282 |jstor=2686282 |archive-url=https://web.archive.org/web/20140629065751/https://www.maa.org/sites/default/files/pdf/pubs/Calc_Articles/ma063.pdf |archive-date=29 June 2014 |url-status=dead}}</ref>

=== Possible side lengths <span class="anchor" id="Inequality"></span> ===
{{main article|Triangle inequality}}
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.<ref>{{multiref
|{{harvnb|Gonick|2024|p=80}}
|{{harvnb|Apostol|1997|p=34–35}}
}}</ref> Conversely, some triangle with three given positive side lengths exists if and only if those side lengths satisfy the triangle inequality.{{sfn|Smith|2000|p=86–87}} The sum of two side lengths can equal the length of the third side only in the case of a [[degenerate triangle]], one with collinear vertices.

=== Rigidity ===
{{main article|Structural rigidity}}
[[File:Structural rigidity basic examples.svg|thumb|Rigidity of a triangle and square]]
Unlike a rectangle, which may collapse into a [[parallelogram]] from pressure to one of its points,{{sfn|Jordan|Smith|2010|p=[https://books.google.com/books?id=tevqDwAAQBAJ&pg=PA834 834]}} triangles are sturdy because specifying the lengths of all three sides determines the angles.{{sfn|Gonick|2024|p=125}} Therefore, 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. A rectangle, in contrast, is more dependent on the strength of its joints in a structural sense.

Triangles are strong in terms of rigidity, but while packed in a [[tessellation|tessellating]] arrangement triangles are not as strong as [[hexagon]]s under compression (hence the prevalence of hexagonal forms in [[nature]]). Tessellated triangles still maintain superior strength for [[cantilever]]ing, however, which is why engineering makes use of [[space frame|tetrahedral trusses]].{{cn|date=August 2024}}

=== Triangulation ===
[[File:Triangulation 3-coloring.svg|thumb|Triangulation in a simple polygon]]
[[Triangulation (geometry)|Triangulation]] means the partition of any planar object into a collection of triangles. For example, in [[polygon triangulation]], a polygon is subdivided into multiple triangles that are attached edge-to-edge, with the property that their vertices coincide with the set of vertices of the polygon.{{sfn|Berg|Kreveld|Overmars|Schwarzkopf|2000}} In the case of a [[simple polygon]] with {{nowrap|1=<math> n </math>}} sides, there are <math> n - 2 </math> triangles that are separated by <math> n - 3 </math> diagonals. Triangulation of a simple polygon has a relationship to the [[Ear (mathematics)|ear]], a vertex connected by two other vertices, the diagonal between which lies entirely within the polygon. The [[two ears theorem]] states that every simple polygon that is not itself a triangle has at least two ears.{{sfn|Meisters|1975}}