Kalman's conjecture: Difference between revisions
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[[File:Tangential trapezoid.svg|thumb|A tangental trapezoid]] | |||
In [[Euclidean geometry]], a '''tangential polygon''', also known as a '''circumscribed polygon''', is a [[convex polygon]] that contains an inscribed circle (also called an ''incircle''). This is a circle that is [[tangent]] to each of the polygon's sides. The [[dual polygon]] of a tangential polygon is a [[cyclic polygon]], which has a circumscribed circle passing through each of its [[vertex (geometry)|vertices]]. | |||
All [[triangle]]s are tangential, as are all [[regular polygon]]s with any number of sides. A well-studied group of tangential polygons are the [[tangential quadrilateral]]s, which include the [[rhombi]]. | |||
==Characterizations== | |||
A convex polygon has an incircle [[if and only if]] all of its internal [[Bisection#Angle bisector|angle bisectors]] are [[Concurrent lines|concurrent]]. This common point is the ''incenter'' (the center of the incircle).<ref>Owen Byer, Felix Lazebnik and Deirdre Smeltzer, ''Methods for Euclidean Geometry'', Mathematical Association of America, 2010, p. 77.</ref> | |||
There exists a tangential polygon of ''n'' sequential sides ''a''<sub>1</sub>, ..., ''a''<sub>''n''</sub> if and only if the [[Simultaneous equations|system of equations]] | |||
:<math>x_1+x_2=a_1,\quad x_2+x_3=a_2,\quad \ldots,\quad x_n+x_1=a_n</math> | |||
has a solution (''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>) in positive [[Real number|reals]].<ref name=IMO/> If such a solution exists, then ''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub> are the ''tangent lengths'' of the polygon (the lengths from the [[Vertex (geometry)|vertices]] to the points where the incircle is [[tangent]] to the sides). | |||
==Inradius== | |||
If the ''n'' sides of a tangential polygon are ''a''<sub>1</sub>, ..., ''a''<sub>''n''</sub>, the inradius ([[radius]] of the incircle) is<ref>Alsina, Claudi and Nelsen, Roger, ''Icons of Mathematics. An exploration of twenty key images'', Mathematical Association of America, 2011, p. 125.</ref> | |||
:<math>r=\frac{K}{s}=\frac{2K}{\sum_{i=1}^n a_i}</math> | |||
where ''K'' is the [[area]] of the polygon and ''s'' is the [[semiperimeter]]. | |||
==Other properties== | |||
*For a tangential polygon with an odd number of sides, all sides are equal if and only if all angles are equal (so the polygon is regular). A tangential polygon with an even number of sides has all sides equal if and only if the alternate angles are equal (that is, angles ''A'', ''C'', ''E'', ... are equal, and angles ''B'', ''D'', ''F'', ... are equal).<ref>De Villiers, Michael. "Equiangular cyclic and equilateral circumscribed polygons," ''[[Mathematical Gazette]]'' 95, March 2011, 102–107.</ref> | |||
*In a tangential polygon with an even number of sides, the sum of the odd numbered sides is equal to the sum of the even numbered sides.<ref name=IMO>Dušan Djukić, Vladimir Janković, Ivan Matić, Nikola Petrović, ''The IMO Compendium'', Springer, 2006, p. 561.</ref> | |||
==Tangential hexagon== | |||
*In a tangential hexagon ''ABCDEF'', the main [[diagonal]]s ''AD'', ''BE'', and ''CF'' are [[Concurrent lines|concurrent]] according to [[Brianchon's theorem]]. | |||
==References== | |||
{{reflist}} | |||
[[Category:Polygons]] | |||
Latest revision as of 08:02, 9 March 2013
In Euclidean geometry, a tangential polygon, also known as a circumscribed polygon, is a convex polygon that contains an inscribed circle (also called an incircle). This is a circle that is tangent to each of the polygon's sides. The dual polygon of a tangential polygon is a cyclic polygon, which has a circumscribed circle passing through each of its vertices.
All triangles are tangential, as are all regular polygons with any number of sides. A well-studied group of tangential polygons are the tangential quadrilaterals, which include the rhombi.
Characterizations
A convex polygon has an incircle if and only if all of its internal angle bisectors are concurrent. This common point is the incenter (the center of the incircle).[1]
There exists a tangential polygon of n sequential sides a1, ..., an if and only if the system of equations
has a solution (x1, ..., xn) in positive reals.[2] If such a solution exists, then x1, ..., xn are the tangent lengths of the polygon (the lengths from the vertices to the points where the incircle is tangent to the sides).
Inradius
If the n sides of a tangential polygon are a1, ..., an, the inradius (radius of the incircle) is[3]
where K is the area of the polygon and s is the semiperimeter.
Other properties
- For a tangential polygon with an odd number of sides, all sides are equal if and only if all angles are equal (so the polygon is regular). A tangential polygon with an even number of sides has all sides equal if and only if the alternate angles are equal (that is, angles A, C, E, ... are equal, and angles B, D, F, ... are equal).[4]
- In a tangential polygon with an even number of sides, the sum of the odd numbered sides is equal to the sum of the even numbered sides.[2]
Tangential hexagon
- In a tangential hexagon ABCDEF, the main diagonals AD, BE, and CF are concurrent according to Brianchon's theorem.
References
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- ↑ Owen Byer, Felix Lazebnik and Deirdre Smeltzer, Methods for Euclidean Geometry, Mathematical Association of America, 2010, p. 77.
- ↑ 2.0 2.1 Dušan Djukić, Vladimir Janković, Ivan Matić, Nikola Petrović, The IMO Compendium, Springer, 2006, p. 561.
- ↑ Alsina, Claudi and Nelsen, Roger, Icons of Mathematics. An exploration of twenty key images, Mathematical Association of America, 2011, p. 125.
- ↑ De Villiers, Michael. "Equiangular cyclic and equilateral circumscribed polygons," Mathematical Gazette 95, March 2011, 102–107.