Editing Triangle (section)

=== 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]}}