En geometría , el círculo o círculo inscrito de un triángulo es el círculo más grande contenido en el triángulo; toca (es tangente a) los tres lados. El centro del círculo es un centro triangular llamado incentro del triángulo . [1]
Un excírculo o círculo escrito [2] del triángulo es un círculo que se encuentra fuera del triángulo, tangente a uno de sus lados y tangente a las extensiones de los otros dos . Cada triángulo tiene tres círculos distintos, cada tangente a uno de los lados del triángulo. [3]
El centro del círculo, llamado incentro , se puede encontrar como la intersección de las tres bisectrices internas del ángulo . [3] [4] El centro de un círculo es la intersección de la bisectriz interna de un ángulo (en el vértice, por ejemplo) y las bisectrices externas de los otros dos. El centro de este círculo se llama excentro en relación con el vértice., o el excenter de. [3] Debido a que la bisectriz interna de un ángulo es perpendicular a su bisectriz externa, se deduce que el centro de la circunferencia junto con los tres centros de la circunferencia forman un sistema ortocéntrico . [5] : pág. 182
Todos los polígonos regulares tienen círculos tangentes a todos los lados, pero no todos los polígonos los tienen; los que lo hacen son polígonos tangenciales . Consulte también Líneas tangentes a círculos .
Circunferencia e incentro
Suponer tiene un círculo con radio y centro . Dejar ser la longitud de , el largo de , y el largo de . También deja, , y ser los puntos de contacto donde el círculo toca , , y .
En el centro
El incentro es el punto donde las bisectrices del ángulo interno de reunirse.
La distancia desde el vértice al incentro es: [ cita requerida ]
Coordenadas trilineales
Las coordenadas trilineales para un punto en el triángulo es la razón de todas las distancias a los lados del triángulo. Debido a que el incentro está a la misma distancia de todos los lados del triángulo, las coordenadas trilineales para el incentro son [6]
Coordenadas baricéntricas
Las coordenadas baricéntricas para un punto en un triángulo dan pesos tales que el punto es el promedio ponderado de las posiciones de los vértices del triángulo. Las coordenadas baricéntricas para el incentro vienen dadas por [ cita requerida ]
dónde , , y son las longitudes de los lados del triángulo, o de manera equivalente (usando la ley de los senos ) por
dónde , , y son los ángulos en los tres vértices.
Coordenadas cartesianas
Las coordenadas cartesianas del incentro son un promedio ponderado de las coordenadas de los tres vértices usando las longitudes de los lados del triángulo en relación con el perímetro (es decir, usando las coordenadas baricéntricas dadas anteriormente, normalizadas para sumar la unidad) como pesos. Los pesos son positivos, por lo que el incentro se encuentra dentro del triángulo como se indicó anteriormente. Si los tres vértices se encuentran en, , y , y los lados opuestos a estos vértices tienen longitudes correspondientes , , y , entonces el incentro está en [ cita requerida ]
Radio
El inradius del círculo en un triángulo con lados de longitud , , viene dado por [7]
- dónde
Vea la fórmula de Heron .
Distancias a los vértices
Denotando el incentro de como , las distancias del incentro a los vértices combinadas con las longitudes de los lados del triángulo obedecen a la ecuación [8]
Además, [9]
dónde y son el circunradio y el radio interno del triángulo, respectivamente.
Otras propiedades
A la colección de centros de triángulos se le puede dar la estructura de un grupo bajo la multiplicación de coordenadas trilineales por coordenadas; en este grupo, el incentro constituye el elemento de identidad . [6]
Incircle y sus propiedades de radio
Distancias entre el vértice y los puntos de contacto más cercanos
Las distancias desde un vértice a los dos puntos de contacto más cercanos son iguales; por ejemplo: [10]
Otras propiedades
Suponga que los puntos de tangencia del círculo dividen los lados en longitudes de y , y , y y . Entonces el círculo tiene el radio [11]
y el área del triángulo es
Si las altitudes de los lados de las longitudes, , y están , , y , luego el inradius es un tercio de la media armónica de estas altitudes; es decir, [12]
El producto del radio de un círculo y la circunferencia circunscrita radio de un triangulo con lados , , y es [5] : 189, # 298 (d)
Some relations among the sides, incircle radius, and circumcircle radius are:[13]
Any line through a triangle that splits both the triangle's area and its perimeter in half goes through the triangle's incenter (the center of its incircle). There are either one, two, or three of these for any given triangle.[14]
Denoting the center of the incircle of as , we have[15]
and[16]:121,#84
The incircle radius is no greater than one-ninth the sum of the altitudes.[17]:289
The squared distance from the incenter to the circumcenter is given by[18]:232
- ,
and the distance from the incenter to the center of the nine point circle is[18]:232
The incenter lies in the medial triangle (whose vertices are the midpoints of the sides).[18]:233, Lemma 1
Relation to area of the triangle
The radius of the incircle is related to the area of the triangle.[19] The ratio of the area of the incircle to the area of the triangle is less than or equal to , with equality holding only for equilateral triangles.[20]
Suppose has an incircle with radius and center . Let be the length of , the length of , and the length of . Now, the incircle is tangent to at some point , and so is right. Thus, the radius is an altitude of . Therefore, has base length and height , and so has area . Similarly, has area and has area . Since these three triangles decompose , we see that the area is:[citation needed]
- and
where is the area of and is its semiperimeter.
For an alternative formula, consider . This is a right-angled triangle with one side equal to and the other side equal to . The same is true for . The large triangle is composed of six such triangles and the total area is:[citation needed]
Gergonne triangle and point
The Gergonne triangle (of ) is defined by the three touchpoints of the incircle on the three sides. The touchpoint opposite is denoted , etc.
This Gergonne triangle, , is also known as the contact triangle or intouch triangle of . Its area is
where , , and are the area, radius of the incircle, and semiperimeter of the original triangle, and , , and are the side lengths of the original triangle. This is the same area as that of the extouch triangle.[21]
The three lines , and intersect in a single point called the Gergonne point, denoted as (or triangle center X7). The Gergonne point lies in the open orthocentroidal disk punctured at its own center, and can be any point therein.[22]
The Gergonne point of a triangle has a number of properties, including that it is the symmedian point of the Gergonne triangle.[23]
Trilinear coordinates for the vertices of the intouch triangle are given by[citation needed]
Trilinear coordinates for the Gergonne point are given by[citation needed]
or, equivalently, by the Law of Sines,
Excircles y excitantes
An excircle or escribed circle[24] of the triangle is a circle lying outside the triangle, tangent to one of its sides and tangent to the extensions of the other two. Every triangle has three distinct excircles, each tangent to one of the triangle's sides.[3]
The center of an excircle is the intersection of the internal bisector of one angle (at vertex , for example) and the external bisectors of the other two. The center of this excircle is called the excenter relative to the vertex , or the excenter of .[3] Because the internal bisector of an angle is perpendicular to its external bisector, it follows that the center of the incircle together with the three excircle centers form an orthocentric system.[5]:182
Trilinear coordinates of excenters
While the incenter of has trilinear coordinates , the excenters have trilinears , , and .[citation needed]
Exradii
The radii of the excircles are called the exradii.
The exradius of the excircle opposite (so touching , centered at ) is[25][26]
- where
See Heron's formula.
Derivation of exradii formula[27]
Let the excircle at side touch at side extended at , and let this excircle's radius be and its center be .
Then is an altitude of , so has area . By a similar argument, has area and has area . Thus the area of triangle is
- .
So, by symmetry, denoting as the radius of the incircle,
- .
By the Law of Cosines, we have
Combining this with the identity , we have
But , and so
which is Heron's formula.
Combining this with , we have
Similarly, gives
and
Other properties
From the formulas above one can see that the excircles are always larger than the incircle and that the largest excircle is the one tangent to the longest side and the smallest excircle is tangent to the shortest side. Further, combining these formulas yields:[28]
Other excircle properties
The circular hull of the excircles is internally tangent to each of the excircles and is thus an Apollonius circle.[29] The radius of this Apollonius circle is where is the incircle radius and is the semiperimeter of the triangle.[30]
The following relations hold among the inradius , the circumradius , the semiperimeter , and the excircle radii , , :[13]
The circle through the centers of the three excircles has radius .[13]
If is the orthocenter of , then[13]
Nagel triangle and Nagel point
The Nagel triangle or extouch triangle of is denoted by the vertices , , and that are the three points where the excircles touch the reference and where is opposite of , etc. This is also known as the extouch triangle of . The circumcircle of the extouch is called the Mandart circle.[citation needed]
The three lines , and are called the splitters of the triangle; they each bisect the perimeter of the triangle,[citation needed]
The splitters intersect in a single point, the triangle's Nagel point (or triangle center X8).
Trilinear coordinates for the vertices of the extouch triangle are given by[citation needed]
Trilinear coordinates for the Nagel point are given by[citation needed]
or, equivalently, by the Law of Sines,
The Nagel point is the isotomic conjugate of the Gergonne point.[citation needed]
Construcciones relacionadas
Nine-point circle and Feuerbach point
In geometry, the nine-point circle is a circle that can be constructed for any given triangle. It is so named because it passes through nine significant concyclic points defined from the triangle. These nine points are:[31][32]
- The midpoint of each side of the triangle
- The foot of each altitude
- The midpoint of the line segment from each vertex of the triangle to the orthocenter (where the three altitudes meet; these line segments lie on their respective altitudes).
In 1822 Karl Feuerbach discovered that any triangle's nine-point circle is externally tangent to that triangle's three excircles and internally tangent to its incircle; this result is known as Feuerbach's theorem. He proved that:[citation needed]
- ... the circle which passes through the feet of the altitudes of a triangle is tangent to all four circles which in turn are tangent to the three sides of the triangle ... ( Feuerbach 1822)
The triangle center at which the incircle and the nine-point circle touch is called the Feuerbach point.
Incentral and excentral triangles
The points of intersection of the interior angle bisectors of with the segments , , and are the vertices of the incentral triangle. Trilinear coordinates for the vertices of the incentral triangle are given by[citation needed]
The excentral triangle of a reference triangle has vertices at the centers of the reference triangle's excircles. Its sides are on the external angle bisectors of the reference triangle (see figure at top of page). Trilinear coordinates for the vertices of the excentral triangle are given by[citation needed]
Ecuaciones para cuatro círculos
Let be a variable point in trilinear coordinates, and let , , . The four circles described above are given equivalently by either of the two given equations:[33]:210–215
- Incircle:
- -excircle:
- -excircle:
- -excircle:
Teorema de euler
Euler's theorem states that in a triangle:
where and are the circumradius and inradius respectively, and is the distance between the circumcenter and the incenter.
For excircles the equation is similar:
where is the radius of one of the excircles, and is the distance between the circumcenter and that excircle's center.[34][35][36]
Generalización a otros polígonos
Some (but not all) quadrilaterals have an incircle. These are called tangential quadrilaterals. Among their many properties perhaps the most important is that their two pairs of opposite sides have equal sums. This is called the Pitot theorem.[citation needed]
More generally, a polygon with any number of sides that has an inscribed circle (that is, one that is tangent to each side) is called a tangential polygon.[citation needed]
Ver también
- Circumgon
- Circumscribed circle – Circle that passes through all the vertices of a polygon
- Ex-tangential quadrilateral
- Harcourt's theorem – Area of a triangle from its sides and vertex distances to any line tangent to its incircle
- Circumconic and inconic – A conic section that passes through the vertices of a triangle or is tangent to its sides
- Inscribed sphere
- Power of a point
- Steiner inellipse
- Tangential quadrilateral
- Trillium theorem – A statement about properties of inscribed and circumscribed circles
Notas
- ^ Kay (1969, p. 140)
- ^ Altshiller-Court (1925, p. 74)
- ^ a b c d e Altshiller-Court (1925, p. 73)
- ^ Kay (1969, p. 117)
- ^ a b c Johnson, Roger A., Advanced Euclidean Geometry, Dover, 2007 (orig. 1929).
- ^ a b Encyclopedia of Triangle Centers Archived 2012-04-19 at the Wayback Machine, accessed 2014-10-28.
- ^ Kay (1969, p. 201)
- ^ Allaire, Patricia R.; Zhou, Junmin; Yao, Haishen (March 2012), "Proving a nineteenth century ellipse identity", Mathematical Gazette, 96: 161–165.
- ^ Altshiller-Court, Nathan (1980), College Geometry, Dover Publications. #84, p. 121.
- ^ Mathematical Gazette, July 2003, 323-324.
- ^ Chu, Thomas, The Pentagon, Spring 2005, p. 45, problem 584.
- ^ Kay (1969, p. 203)
- ^ a b c d Bell, Amy, "Hansen’s right triangle theorem, its converse and a generalization", Forum Geometricorum 6, 2006, 335–342.
- ^ Kodokostas, Dimitrios, "Triangle Equalizers," Mathematics Magazine 83, April 2010, pp. 141-146.
- ^ Allaire, Patricia R.; Zhou, Junmin; and Yao, Haishen, "Proving a nineteenth century ellipse identity", Mathematical Gazette 96, March 2012, 161-165.
- ^ Altshiller-Court, Nathan. College Geometry, Dover Publications, 1980.
- ^ Posamentier, Alfred S., and Lehmann, Ingmar. The Secrets of Triangles, Prometheus Books, 2012.
- ^ a b c Franzsen, William N. (2011). "The distance from the incenter to the Euler line" (PDF). Forum Geometricorum. 11: 231–236. MR 2877263..
- ^ Coxeter, H.S.M. "Introduction to Geometry 2nd ed. Wiley, 1961.
- ^ Minda, D., and Phelps, S., "Triangles, ellipses, and cubic polynomials", American Mathematical Monthly 115, October 2008, 679-689: Theorem 4.1.
- ^ Weisstein, Eric W. "Contact Triangle." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/ContactTriangle.html
- ^ Christopher J. Bradley and Geoff C. Smith, "The locations of triangle centers", Forum Geometricorum 6 (2006), 57–70. http://forumgeom.fau.edu/FG2006volume6/FG200607index.html
- ^ Dekov, Deko (2009). "Computer-generated Mathematics : The Gergonne Point" (PDF). Journal of Computer-generated Euclidean Geometry. 1: 1–14. Archived from the original (PDF) on 2010-11-05.
- ^ Altshiller-Court (1925, p. 74)
- ^ Altshiller-Court (1925, p. 79)
- ^ Kay (1969, p. 202)
- ^ Altshiller-Court (1925, p. 79)
- ^ Baker, Marcus, "A collection of formulae for the area of a plane triangle," Annals of Mathematics, part 1 in vol. 1(6), January 1885, 134-138. (See also part 2 in vol. 2(1), September 1885, 11-18.)
- ^ Grinberg, Darij, and Yiu, Paul, "The Apollonius Circle as a Tucker Circle", Forum Geometricorum 2, 2002: pp. 175-182.
- ^ Stevanovi´c, Milorad R., "The Apollonius circle and related triangle centers", Forum Geometricorum 3, 2003, 187-195.
- ^ Altshiller-Court (1925, pp. 103–110)
- ^ Kay (1969, pp. 18,245)
- ^ Whitworth, William Allen. Trilinear Coordinates and Other Methods of Modern Analytical Geometry of Two Dimensions, Forgotten Books, 2012 (orig. Deighton, Bell, and Co., 1866). http://www.forgottenbooks.com/search?q=Trilinear+coordinates&t=books
- ^ Nelson, Roger, "Euler's triangle inequality via proof without words," Mathematics Magazine 81(1), February 2008, 58-61.
- ^ Johnson, R. A. Modern Geometry, Houghton Mifflin, Boston, 1929: p. 187.
- ^ Emelyanov, Lev, and Emelyanova, Tatiana. "Euler’s formula and Poncelet’s porism", Forum Geometricorum 1, 2001: pp. 137–140.
Referencias
- Altshiller-Court, Nathan (1925), College Geometry: An Introduction to the Modern Geometry of the Triangle and the Circle (2nd ed.), New York: Barnes & Noble, LCCN 52013504
- Kay, David C. (1969), College Geometry, New York: Holt, Rinehart and Winston, LCCN 69012075
- Kimberling, Clark (1998). "Triangle Centers and Central Triangles". Congressus Numerantium (129): i–xxv, 1–295.
- Kiss, Sándor (2006). "The Orthic-of-Intouch and Intouch-of-Orthic Triangles". Forum Geometricorum (6): 171–177.
enlaces externos
- Derivation of formula for radius of incircle of a triangle
- Weisstein, Eric W. "Incircle". MathWorld.
Interactive
- Triangle incenter Triangle incircle Incircle of a regular polygon With interactive animations
- Constructing a triangle's incenter / incircle with compass and straightedge An interactive animated demonstration
- Equal Incircles Theorem at cut-the-knot
- Five Incircles Theorem at cut-the-knot
- Pairs of Incircles in a Quadrilateral at cut-the-knot
- An interactive Java applet for the incenter