Editing Triangle (section)

=== 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]]
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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}}