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Tangential polygon
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<h2 id="characterizations">Characterizations</h2> <p>A convex polygon has an incircle <a href="/facts/If_and_only_if/bYSxGJ66">if and only if</a> all of its internal <a href="/facts/Bisection/JTsya7Ca">angle bisectors</a> are <a href="/facts/Concurrent_lines/zPT6S4w9">concurrent</a>. This common point is the <i>incenter</i> (the center of the incircle).<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> </p><p>There exists a tangential polygon of <i>n</i> sequential sides <i>a</i>1, ..., <i>a</i><i>n</i> if and only if the <a href="/facts/Simultaneous_equations/KzP6TJ4u">system of equations</a> </p> x 1 + x 2 = a 1 , x 2 + x 3 = a 2 , … , x n + x 1 = a n {\displaystyle x_{1}+x_{2}=a_{1},\quad x_{2}+x_{3}=a_{2},\quad \ldots ,\quad x_{n}+x_{1}=a_{n}} <p>has a solution (<i>x</i>1, ..., <i>x</i><i>n</i>) in positive <a href="/facts/Real_number/R02gw5Pb">reals</a>.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> If such a solution exists, then <i>x</i>1, ..., <i>x</i><i>n</i> are the <i>tangent lengths</i> of the polygon (the lengths from the <a href="/facts/Vertex_(geometry)/ID3cea9z">vertices</a> to the points where the incircle is <a href="/facts/Tangent/sF0jH3EI">tangent</a> to the sides). </p> <h2 id="uniqueness-and-non-uniqueness">Uniqueness and non-uniqueness</h2> <p>If the number of sides <i>n</i> is odd, then for any given set of sidelengths a 1 , … , a n {\displaystyle a_{1},\dots ,a_{n}} satisfying the existence criterion above there is only one tangential polygon. But if <i>n</i> is even there are an infinitude of them.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a>: p. 389 For example, in the quadrilateral case where all sides are equal we can have a <a href="/facts/Rhombus/T4pU50GF">rhombus</a> with any value of the acute angles, and all rhombi are tangential to an incircle. </p> <h2 id="inradius">Inradius</h2> <p>If the <i>n</i> sides of a tangential polygon are <i>a</i>1, ..., <i>a</i><i>n</i>, the inradius (<a href="/facts/Radius/9Gr9deYD">radius</a> of the incircle) is<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> </p> r = K s = 2 K ∑ i = 1 n a i {\displaystyle r={\frac {K}{s}}={\frac {2K}{\sum _{i=1}^{n}a_{i}}}} <p>where <i>K</i> is the <a href="/facts/Area/KsNG1G9t">area</a> of the polygon and <i>s</i> is the <a href="/facts/Semiperimeter/8AqgmdqM">semiperimeter</a>. (Since all <a href="/facts/Triangle/cRXUwsMn">triangles</a> are tangential, this formula applies to all triangles.) </p> <h2 id="other-properties">Other properties</h2> <ul><li>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 <i>A</i>, <i>C</i>, <i>E</i>, ... are equal, and angles <i>B</i>, <i>D</i>, <i>F</i>, ... are equal).<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a></li> <li>In a tangential polygon with an even number of sides, the sum of the odd numbered sides' lengths is equal to the sum of the even numbered sides' lengths.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a></li> <li>A tangential polygon has a larger area than any other polygon with the same perimeter and the same interior angles in the same sequence.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a>: p. 862 <a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a></li> <li>The <a href="/facts/Centroid/H6dDbg0h">centroid</a> of any tangential polygon, the centroid of its boundary points, and the center of the inscribed circle are <a href="/facts/Collinearity/SxbVBJYR">collinear</a>, with the polygon's centroid between the others and twice as far from the incenter as from the boundary's centroid.<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a>: pp. 858–9 </li></ul> <h2 id="tangential-triangle">Tangential triangle</h2> <p>While all triangles are tangential to some circle, a triangle is called the <a href="/facts/Tangential_triangle/5r8CTdsC">tangential triangle</a> of a reference triangle if the tangencies of the tangential triangle with the circle are also the vertices of the reference triangle. </p> <h2 id="tangential-quadrilateral">Tangential quadrilateral</h2> <p class="note">Main article: <a href="/facts/Tangential_quadrilateral/JBdfotIx">Tangential quadrilateral</a></p> <h2 id="tangential-hexagon">Tangential hexagon</h2> <ul><li>In a tangential <a href="/facts/Hexagon/DPzuCzWE">hexagon</a> <i>ABCDEF</i>, the main <a href="/facts/Diagonal/39z9sIRb">diagonals</a> <i>AD</i>, <i>BE</i>, and <i>CF</i> are <a href="/facts/Concurrent_lines/zPT6S4w9">concurrent</a> according to <a href="/facts/Brianchon%2527s_theorem/pqHflumv">Brianchon's theorem</a>.</li></ul> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Circumgon/zWsLu5EB">Circumgon</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Owen Byer, Felix Lazebnik and Deirdre Smeltzer, Methods for Euclidean Geometry, Mathematical Association of America, 2010, p. 77. <a href="/wiki/Deirdre_Smeltzer" target="_blank">/wiki/Deirdre_Smeltzer</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Dušan Djukić, Vladimir Janković, Ivan Matić, Nikola Petrović, The IMO Compendium, Springer, 2006, p. 561. <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Hess, Albrecht (2014), "On a circle containing the incenters of tangential quadrilaterals" (PDF), Forum Geometricorum, 14: 389–396. <a href="http://forumgeom.fau.edu/FG2014volume14/FG201437.pdf" target="_blank">http://forumgeom.fau.edu/FG2014volume14/FG201437.pdf</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Alsina, Claudi and Nelsen, Roger, Icons of Mathematics. An exploration of twenty key images, Mathematical Association of America, 2011, p. 125. <a href="/wiki/Icons_of_Mathematics" target="_blank">/wiki/Icons_of_Mathematics</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>De Villiers, Michael. "Equiangular cyclic and equilateral circumscribed polygons," Mathematical Gazette 95, March 2011, 102–107. <a href="/wiki/Mathematical_Gazette" target="_blank">/wiki/Mathematical_Gazette</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Dušan Djukić, Vladimir Janković, Ivan Matić, Nikola Petrović, The IMO Compendium, Springer, 2006, p. 561. <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Tom M. Apostol and Mamikon A. Mnatsakanian (December 2004). "Figures Circumscribing Circles" (PDF). American Mathematical Monthly. 111 (10): 853–863. doi:10.2307/4145094. JSTOR 4145094. Retrieved 6 April 2016. <a href="http://www.maa.org/sites/default/files/images/upload_library/22/Ford/Apostol853-863.pdf" target="_blank">http://www.maa.org/sites/default/files/images/upload_library/22/Ford/Apostol853-863.pdf</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Apostol, Tom (December 2005). "erratum". American Mathematical Monthly. 112 (10): 946. doi:10.1080/00029890.2005.11920274. S2CID 218547110. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Tom M. Apostol and Mamikon A. Mnatsakanian (December 2004). "Figures Circumscribing Circles" (PDF). American Mathematical Monthly. 111 (10): 853–863. doi:10.2307/4145094. JSTOR 4145094. Retrieved 6 April 2016. <a href="http://www.maa.org/sites/default/files/images/upload_library/22/Ford/Apostol853-863.pdf" target="_blank">http://www.maa.org/sites/default/files/images/upload_library/22/Ford/Apostol853-863.pdf</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> </ol>