Ex-tangential quadrilateral

<h2 id="special-cases">Special cases</h2>
<a href="/facts/Kite_(geometry)/PRcvPntl">Kites</a> are examples of ex-tangential quadrilaterals. <a href="/facts/Parallelograms/DRuktq60">Parallelograms</a> (which include <a href="/facts/Square_(geometry)/JHJeMvmL">squares</a>, <a href="/facts/Rhombus/T4pU50GF">rhombi</a>, and <a href="/facts/Rectangle/mdLfIGKA">rectangles</a>) can be considered as a <a href="/facts/Degeneracy_(mathematics)/BqhqlXPH">degenerate</a> case of ex-tangential quadrilaterals with <a href="/facts/Infinity/cF506uD7">infinite</a> exradius since they satisfy the characterizations in the next section, but the excircle cannot be tangent to both pairs of extensions of opposite sides (since they are parallel).<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a> Convex quadrilaterals whose side lengths form an <a href="/facts/Arithmetic_progression/c4ew9Xoq">arithmetic progression</a> are always ex-tangential as they satisfy the characterization below for adjacent side lengths.

<h2 id="characterizations">Characterizations</h2>
A convex quadrilateral is ex-tangential <a href="/facts/If_and_only_if/bYSxGJ66">if and only if</a> there are six <a href="/facts/Concurrent_lines/zPT6S4w9">concurrent</a> angles bisectors. These are the internal <a href="/facts/Bisection/JTsya7Ca">angle bisectors</a> at two opposite vertex angles, the external angle bisectors at the other two vertex
angles, and the external angle bisectors at the angles formed where the extensions of opposite sides intersect.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a>
For the purpose of calculation, a more useful characterization is that a convex quadrilateral with successive sides a, b, c, d is ex-tangential if and only if the sum of two adjacent sides is equal to the sum of the other two sides. This is possible in two different ways:

a
        +
        b
        =
        c
        +
        d
      
    
    {\displaystyle a+b=c+d}

or

a
        +
        d
        =
        b
        +
        c
        .
      
    
    {\displaystyle a+d=b+c.}

This was proved by <a href="/facts/Jakob_Steiner/cXzHkeYt">Jakob Steiner</a> in 1846.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a> In the first case, the excircle is outside the biggest of the vertices A or C, whereas in the second case it is outside the biggest of the vertices B or D, provided that the sides of the quadrilateral ABCD are

a
        =
        
          |
        
        A
        B
        
          |
        
        ,
         
        b
        =
        
          |
        
        B
        C
        
          |
        
        ,
         
        c
        =
        
          |
        
        C
        D
        
          |
        
        ,
         
        d
        =
        
          |
        
        D
        A
        
          |
        
        .
      
    
    {\displaystyle a=|AB|,\ b=|BC|,\ c=|CD|,\ d=|DA|.}

A way of combining these characterizations regarding the sides is that the <a href="/facts/Absolute_value/dwJBpEs5">absolute values</a> of the differences between opposite sides are equal for the two pairs of opposite sides,<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a>

|
        
        a
        −
        c
        
          |
        
        =
        
          |
        
        b
        −
        d
        
          |
        
        .
      
    
    {\displaystyle |a-c|=|b-d|.}

These equations are closely related to the <a href="/facts/Pitot_theorem/9Ttd6uAD">Pitot theorem</a> for <a href="/facts/Tangential_quadrilateral/JBdfotIx">tangential quadrilaterals</a>, where the sums of opposite sides are equal for the two pairs of opposite sides.

<h3>Urquhart's theorem</h3>
If opposite sides in a convex quadrilateral ABCD intersect at E and F, then

|
        
        A
        B
        
          |
        
        +
        
          |
        
        B
        C
        
          |
        
        =
        
          |
        
        A
        D
        
          |
        
        +
        
          |
        
        D
        C
        
          |
        
        
        ⇔
        
        
          |
        
        A
        E
        
          |
        
        +
        
          |
        
        E
        C
        
          |
        
        =
        
          |
        
        A
        F
        
          |
        
        +
        
          |
        
        F
        C
        
          |
        
        .
      
    
    {\displaystyle |AB|+|BC|=|AD|+|DC|\quad \Leftrightarrow \quad |AE|+|EC|=|AF|+|FC|.}

The implication to the right is named after L. M. Urquhart (1902–1966) although it was proved long before by <a href="/facts/Augustus_De_Morgan/6fQ1KTPb">Augustus De Morgan</a> in 1841. <a href="/facts/Daniel_Pedoe/uVGSaTQU">Daniel Pedoe</a> named it the most elementary theorem in <a href="/facts/Euclidean_geometry/V6cTcoCy">Euclidean geometry</a> since it only concerns straight lines and distances.<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a> That there in fact is an equivalence was proved by Mowaffac Hajja,<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a> which makes the equality to the right another <a href="/facts/Necessary_and_sufficient_condition/G45aDCY9">necessary and sufficient condition</a> for a quadrilateral to be ex-tangential.

<h3>Comparison with a tangential quadrilateral</h3>
A few of the metric characterizations of <a href="/facts/Tangential_quadrilateral/JBdfotIx">tangential quadrilaterals</a> (the left column in the table) have very similar counterparts for ex-tangential quadrilaterals (the middle and right column in the table), as can be seen in the table below.<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a> Thus a convex quadrilateral has an incircle or an excircle outside the appropriate vertex (depending on the column) if and only if any one of the five necessary and sufficient conditions below is satisfied.

<table><tbody><tr><th>Incircle</th><th>Excircle outside of A or C</th><th>Excircle outside of B or D</th></tr><tr><td> R 1 + R 3 = R 2 + R 4 {\displaystyle R_{1}+R_{3}=R_{2}+R_{4}} </td><td> R 1 + R 2 = R 3 + R 4 {\displaystyle R_{1}+R_{2}=R_{3}+R_{4}} </td><td> R 1 + R 4 = R 2 + R 3 {\displaystyle R_{1}+R_{4}=R_{2}+R_{3}} </td></tr><tr><td> a g h + c e f = b e h + d f g {\displaystyle agh+cef=beh+dfg} </td><td> a g h + b e h = c e f + d f g {\displaystyle agh+beh=cef+dfg} </td><td> a g h + d f g = b e h + c e f {\displaystyle agh+dfg=beh+cef} </td></tr><tr><td> 1 h 1 + 1 h 3 = 1 h 2 + 1 h 4 {\displaystyle {\frac {1}{h_{1}}}+{\frac {1}{h_{3}}}={\frac {1}{h_{2}}}+{\frac {1}{h_{4}}}} </td><td> 1 h 1 + 1 h 2 = 1 h 3 + 1 h 4 {\displaystyle {\frac {1}{h_{1}}}+{\frac {1}{h_{2}}}={\frac {1}{h_{3}}}+{\frac {1}{h_{4}}}} </td><td> 1 h 1 + 1 h 4 = 1 h 2 + 1 h 3 {\displaystyle {\frac {1}{h_{1}}}+{\frac {1}{h_{4}}}={\frac {1}{h_{2}}}+{\frac {1}{h_{3}}}} </td></tr><tr><td> tan ⁡ x 2 tan ⁡ z 2 = tan ⁡ y 2 tan ⁡ w 2 {\displaystyle \tan {\frac {x}{2}}\tan {\frac {z}{2}}=\tan {\frac {y}{2}}\tan {\frac {w}{2}}} </td><td> tan ⁡ x 2 tan ⁡ w 2 = tan ⁡ y 2 tan ⁡ z 2 {\displaystyle \tan {\frac {x}{2}}\tan {\frac {w}{2}}=\tan {\frac {y}{2}}\tan {\frac {z}{2}}} </td><td> tan ⁡ x 2 tan ⁡ y 2 = tan ⁡ z 2 tan ⁡ w 2 {\displaystyle \tan {\frac {x}{2}}\tan {\frac {y}{2}}=\tan {\frac {z}{2}}\tan {\frac {w}{2}}} </td></tr><tr><td> R a R c = R b R d {\displaystyle R_{a}R_{c}=R_{b}R_{d}} </td><td> R a R b = R c R d {\displaystyle R_{a}R_{b}=R_{c}R_{d}} </td><td> R a R d = R b R c {\displaystyle R_{a}R_{d}=R_{b}R_{c}} </td></tr></tbody></table>
The notations in this table are as follows: In a convex quadrilateral ABCD, the diagonals intersect at P. 

<ul><li>R1, R2, R3, R4 are the circumradii in triangles △ABP, △BCP, △CDP, △DAP;</li>
<li>h1, h2, h3, h4 are the altitudes from P to the sides a = |AB|, b = |BC|, c = |CD|, d = |DA| respectively in the same four triangles;</li>
<li>e, f, g, h are the distances from the vertices A, B, C, D respectively to P;</li>
<li>x, y, z, w are the angles ∠ABD, ∠ADB, ∠BDC, ∠DBC respectively;</li>
<li>and Ra, Rb, Rc, Rd are the radii in the circles externally tangent to the sides a, b, c, d respectively and the extensions of the adjacent two sides for each side.</li></ul>
<h2 id="area">Area</h2>
An ex-tangential quadrilateral ABCD with sides a, b, c, d has area

K
          =
          
            
              a
              b
              c
              d
            
          
          sin
          ⁡
          
            
              
                B
                +
                D
              
              2
            
          
          .
        
      
    
    {\displaystyle \displaystyle K={\sqrt {abcd}}\sin {\frac {B+D}{2}}.}

Note that this is the same formula as the one for the area of a <a href="/facts/Tangential_quadrilateral/JBdfotIx">tangential quadrilateral</a> and it is also derived from <a href="/facts/Bretschneider%2527s_formula/0EPJIxyx">Bretschneider's formula</a> in the same way.

<h2 id="exradius">Exradius</h2>
The exradius for an ex-tangential quadrilateral with consecutive sides a, b, c, d is given by<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a>

r
        =
        
          
            K
            
              
                |
              
              a
              −
              c
              
                |
              
            
          
        
        =
        
          
            K
            
              
                |
              
              b
              −
              d
              
                |
              
            
          
        
      
    
    {\displaystyle r={\frac {K}{|a-c|}}={\frac {K}{|b-d|}}}

where K is the area of the quadrilateral. For an ex-tangential quadrilateral with given sides, the exradius is <a href="/facts/Maxima_and_minima/ADlgnyzV">maximum</a> when the quadrilateral is also <a href="/facts/Cyclic_quadrilateral/FNwQ0cda">cyclic</a> (and hence an ex-bicentric quadrilateral). These formulas explain why all parallelograms have infinite exradius.

<h2 id="ex-bicentric-quadrilateral">Ex-bicentric quadrilateral</h2>
If an ex-tangential quadrilateral also has a <a href="/facts/Circumcircle/RafYT4am">circumcircle</a>, it is called an ex-bicentric quadrilateral.<a class="footnote-ref" id="fnref:13" href="#fn:13">13</a> Then, since it has two opposite <a href="/facts/Supplementary_angles/gFUTEQYq">supplementary angles</a>, its area is given by

K
          =
          
            
              a
              b
              c
              d
            
          
        
      
    
    {\displaystyle \displaystyle K={\sqrt {abcd}}}

which is the same as for a <a href="/facts/Bicentric_quadrilateral/k5BWD6Bm">bicentric quadrilateral</a>.
If x is the distance between the <a href="/facts/Circumcenter/RafYT4am">circumcenter</a> and the excenter, then<a class="footnote-ref" id="fnref:14" href="#fn:14">14</a>

1
            
              (
              R
              −
              x
              
                )
                
                  2
                
              
            
          
        
        +
        
          
            1
            
              (
              R
              +
              x
              
                )
                
                  2
                
              
            
          
        
        =
        
          
            1
            
              r
              
                2
              
            
          
        
        ,
      
    
    {\displaystyle {\frac {1}{(R-x)^{2}}}+{\frac {1}{(R+x)^{2}}}={\frac {1}{r^{2}}},}

where R, r are the <a href="/facts/Circumradius/RafYT4am">circumradius</a> and exradius respectively. This is the same equation as <a href="/facts/Bicentric_quadrilateral/k5BWD6Bm">Fuss's theorem</a> for a bicentric quadrilateral. But when solving for x, we must choose the other root of the <a href="/facts/Quadratic_equation/9IMnkJAT">quadratic equation</a> for the ex-bicentric quadrilateral compared to the bicentric. Hence, for the ex-bicentric we have<a class="footnote-ref" id="fnref:15" href="#fn:15">15</a>

x
        =
        
          
            
              R
              
                2
              
            
            +
            
              r
              
                2
              
            
            +
            r
            
              
                4
                
                  R
                  
                    2
                  
                
                +
                
                  r
                  
                    2
                  
                
              
            
          
        
        .
      
    
    {\displaystyle x={\sqrt {R^{2}+r^{2}+r{\sqrt {4R^{2}+r^{2}}}}}.}

From this formula it follows that

x
          >
          R
          +
          r
          ,
        
      
    
    {\displaystyle \displaystyle x>R+r,}

which means that the circumcircle and excircle can never intersect each other.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Complete_quadrangle/GWuCsbYm">Complete quadrangle</a></li>
<li><a href="/facts/Cyclic_quadrilateral/FNwQ0cda">Cyclic quadrilateral</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Radic, Mirko; Kaliman, Zoran and Kadum, Vladimir, "A condition that a tangential quadrilateral is also a chordal one", Mathematical Communications, 12 (2007) pp. 33–52. <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Bogomolny, Alexander, "Inscriptible and Exscriptible Quadrilaterals", Interactive Mathematics Miscellany and Puzzles, [1]. Accessed 2011-08-18. <a href="/wiki/Alexander_Bogomolny" target="_blank">/wiki/Alexander_Bogomolny</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">K. S. Kedlaya, Geometry Unbound, 2006 <a href="https://kskedlaya.org/geometryunbound/gu-060118.pdf" target="_blank">https://kskedlaya.org/geometryunbound/gu-060118.pdf</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Josefsson, Martin, Similar Metric Characterizations of Tangential and Extangential Quadrilaterals, Forum Geometricorum Volume 12 (2012) pp. 63-77 [2] Archived 2022-01-16 at the Wayback Machine <a href="http://forumgeom.fau.edu/FG2012volume12/FG201207.pdf" target="_blank">http://forumgeom.fau.edu/FG2012volume12/FG201207.pdf</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Josefsson, Martin, Similar Metric Characterizations of Tangential and Extangential Quadrilaterals, Forum Geometricorum Volume 12 (2012) pp. 63-77 [2] Archived 2022-01-16 at the Wayback Machine <a href="http://forumgeom.fau.edu/FG2012volume12/FG201207.pdf" target="_blank">http://forumgeom.fau.edu/FG2012volume12/FG201207.pdf</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Josefsson, Martin, Similar Metric Characterizations of Tangential and Extangential Quadrilaterals, Forum Geometricorum Volume 12 (2012) pp. 63-77 [2] Archived 2022-01-16 at the Wayback Machine <a href="http://forumgeom.fau.edu/FG2012volume12/FG201207.pdf" target="_blank">http://forumgeom.fau.edu/FG2012volume12/FG201207.pdf</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">F. G.-M., Exercices de Géométrie, Éditions Jacques Gabay, sixiéme édition, 1991, p. 318. <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Josefsson, Martin, Similar Metric Characterizations of Tangential and Extangential Quadrilaterals, Forum Geometricorum Volume 12 (2012) pp. 63-77 [2] Archived 2022-01-16 at the Wayback Machine <a href="http://forumgeom.fau.edu/FG2012volume12/FG201207.pdf" target="_blank">http://forumgeom.fau.edu/FG2012volume12/FG201207.pdf</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Hajja, Mowaffaq, A Very Short and Simple Proof of “The Most Elementary Theorem” of Euclidean Geometry, Forum Geometricorum Volume 6 (2006) pp. 167–169 [3] Archived 2021-11-19 at the Wayback Machine <a href="http://forumgeom.fau.edu/FG2006volume6/FG200616.pdf" target="_blank">http://forumgeom.fau.edu/FG2006volume6/FG200616.pdf</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Hajja, Mowaffaq, A Very Short and Simple Proof of “The Most Elementary Theorem” of Euclidean Geometry, Forum Geometricorum Volume 6 (2006) pp. 167–169 [3] Archived 2021-11-19 at the Wayback Machine <a href="http://forumgeom.fau.edu/FG2006volume6/FG200616.pdf" target="_blank">http://forumgeom.fau.edu/FG2006volume6/FG200616.pdf</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Josefsson, Martin, Similar Metric Characterizations of Tangential and Extangential Quadrilaterals, Forum Geometricorum Volume 12 (2012) pp. 63-77 [2] Archived 2022-01-16 at the Wayback Machine <a href="http://forumgeom.fau.edu/FG2012volume12/FG201207.pdf" target="_blank">http://forumgeom.fau.edu/FG2012volume12/FG201207.pdf</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">Josefsson, Martin, Similar Metric Characterizations of Tangential and Extangential Quadrilaterals, Forum Geometricorum Volume 12 (2012) pp. 63-77 [2] Archived 2022-01-16 at the Wayback Machine <a href="http://forumgeom.fau.edu/FG2012volume12/FG201207.pdf" target="_blank">http://forumgeom.fau.edu/FG2012volume12/FG201207.pdf</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">Radic, Mirko; Kaliman, Zoran and Kadum, Vladimir, "A condition that a tangential quadrilateral is also a chordal one", Mathematical Communications, 12 (2007) pp. 33–52. <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">Radic, Mirko; Kaliman, Zoran and Kadum, Vladimir, "A condition that a tangential quadrilateral is also a chordal one", Mathematical Communications, 12 (2007) pp. 33–52. <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
<li id="fn:15">Radic, Mirko; Kaliman, Zoran and Kadum, Vladimir, "A condition that a tangential quadrilateral is also a chordal one", Mathematical Communications, 12 (2007) pp. 33–52. <a href="#fnref:15" class="footnote-back-ref">↩</a></li>
</ol>

Ex-tangential quadrilateral open-in-new

Ex-tangential quadrilateral