Editing Triangle (section)

== Related figures ==
=== Figures inscribed in a 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 [[Ellipse#Elements of an ellipse|foci of this ellipse]].{{sfn|Kalman|2008}} 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 <math> ABC </math>, let the foci be <math> P </math> and <math> Q </math>, then:{{sfn|Allaire|Zhou|Yao|2012}}
<math display="block"> \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. </math>

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 | image1 = Pedal Triangle.svg
 | image2 = Intouch Triangle and Gergonne Point.svg
 | footer = The [[pedal triangle]] and [[Gergonne triangle]]
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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.{{sfn|Coxeter|Greitzer|1967|pp=18,23–25}}

The [[intouch triangle]] of a reference triangle has its vertices at the three points of tangency of the reference triangle's sides with its incircle.<ref>{{Cite journal |last=Kimberling |first=Clark |date=March 2008 |title=Twenty-one points on the nine-point circle |url=https://www.cambridge.org/core/product/identifier/S002555720018249X/type/journal_article |journal=The Mathematical Gazette |language=en |volume=92 |issue=523 |pages=29–38 |doi=10.1017/S002555720018249X |issn=0025-5572}}</ref> 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).<ref>{{Cite journal |last1=Moses |first1=Peter |last2=Kimberling |first2=Charles |date=2009 |title=Reflection-Induced Perspectivities Among Triangles |url=https://www.heldermann-verlag.de/jgg/jgg13/j13h1mose.pdf |journal=Journal for Geometry and Graphics |volume=13 |issue=1 |pages=15–24}}</ref>

[[File:Calabi triangle.svg|thumb|The [[Calabi triangle]] and the three placements of its largest square. The placement on the long side of the triangle is inscribed; the other two are not.]]
[[Inscribed square in a triangle|The inscribed squares tangent their vertices to the triangle's sides]] is the special case of [[inscribed square problem]], although the problem asking for a square whose vertices lie on a [[simple closed curve]]. A notable example of this figure relation is the [[Calabi triangle]] in which the vertices of every three squares are tangent to all obtuse triangle's sides. 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 a side of length <math> q_a </math> and the triangle has a side of length <math> a </math>, part of which side coincides with a side of the square, then <math> q_a </math>, <math> a </math>, <math> h_a </math> from the side <math> a </math>, and the triangle's area <math> T </math> are related according to<ref>{{multiref
|{{harvnb|Bailey|Detemple|1998}}
|{{harvnb|Oxman|Stupel|2013}}
}}</ref><math display="block"> q_a=\frac{2Ta}{a^2+2T} = \frac{ah_a}{a+h_a}. </math>The largest possible ratio of the area of the inscribed square to the area of the triangle is 1/2, which occurs when <math> a^2 = 2T </math>, <math> q = a/2 </math>, and the altitude of the triangle from the base of length <math> a </math> is equal to <math> a </math>. The smallest possible ratio of the side of one inscribed square to the side of another in the same non-obtuse triangle is <math>2\sqrt{2}/3</math>.{{sfn|Oxman|Stupel|2013}} Both of these extreme cases occur for the isosceles right triangle.{{cn|date=August 2024}}

[[File:Lemoine Hexagon.svg|thumb|The Lemoine hexagon inscribed in a triangle]]
The [[Lemoine hexagon]] is a [[hexagon#Cyclic hexagon|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 [[polygon#Convexity and types of non-convexity|simple form or its self-intersecting form]], the Lemoine hexagon is interior to the triangle with two vertices on each side of the triangle.{{cn|date=August 2024}}

Every [[convex polygon]] with area <math> T </math> can be inscribed in a triangle of area at most equal to <math> 2T </math>. Equality holds only if the polygon is a [[parallelogram]].{{sfn|Eggleston|2007|pp=149–160}}

=== Figures circumscribed about a triangle ===
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 | image1 = Tangential triangle.svg
 | image2 = Steiner ellipse.svg
 | footer = The circumscribed circle tangent to a triangle and the [[Steiner circumellipse]]
}}
The [[tangential triangle]] of a reference triangle (other than a right triangle) is the triangle whose sides are on the [[tangent line]]s to the reference triangle's circumcircle at its vertices.<ref>{{cite journal|last1=Smith |first1=Geoff |last2=Leversha |first2=Gerry |title=Euler and triangle geometry |journal=Mathematical Gazette |volume=91 |date=November 2007 |issue=522 |pages=436–452 |doi=10.1017/S0025557200182087 |jstor=40378417}}</ref>

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. Furthermore, every triangle has a unique [[Steiner ellipse|Steiner circumellipse]], which passes through the triangle's vertices and has its center at the triangle's centroid. Of all ellipses going through the triangle's vertices, it has the smallest area.<ref>{{cite journal|first=John R. |last=Silvester |title=Extremal area ellipses of a convex quadrilateral |journal=The Mathematical Gazette |volume=101 |number=550 |date=March 2017 |pages=11–26 |doi=10.1017/mag.2017.2 }}</ref>

The [[Kiepert hyperbola]] is unique [[conic]] that passes through the triangle's three vertices, its centroid, and its circumcenter.<ref>{{cite journal |last1=Eddy |first1=R. H. |last2=Fritsch |first2=R. |title=The Conics of Ludwig Kiepert: A Comprehensive Lesson in the Geometry of the Triangle |journal=Mathematics Magazine |date=1994 |volume=67 |issue=3 |pages=188–205|doi=10.1080/0025570X.1994.11996212 }}</ref>

Of all triangles contained in a given [[convex polygon]], one with maximal area can be found in linear time; its vertices may be chosen as three of the vertices of the given polygon.{{sfn|Chandran|Mount|1992}}