Circumconic and inconic

<h2 id="centers-and-tangent-lines">Centers and tangent lines</h2>
<h3>Circumconic</h3>
<p>The center of the general circumconic is the point
</p>

u
        (
        −
        a
        u
        +
        b
        v
        +
        c
        w
        )
        :
        v
        (
        a
        u
        −
        b
        v
        +
        c
        w
        )
        :
        w
        (
        a
        u
        +
        b
        v
        −
        c
        w
        )
        .
      
    
    {\displaystyle u(-au+bv+cw):v(au-bv+cw):w(au+bv-cw).}

<p>The lines tangent to the general circumconic at the vertices A, B, C are, respectively,
</p>

w
                v
                +
                v
                z
              
              
                
                =
                0
                ,
              
            
            
              
                u
                z
                +
                w
                x
              
              
                
                =
                0
                ,
              
            
            
              
                v
                x
                +
                u
                y
              
              
                
                =
                0.
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}wv+vz&=0,\\uz+wx&=0,\\vx+uy&=0.\end{aligned}}}

<h3>Inconic</h3>
<p>The center of the general inconic is the point
</p>

c
        v
        +
        b
        w
        :
        a
        w
        +
        c
        u
        :
        b
        u
        +
        a
        v
        .
      
    
    {\displaystyle cv+bw:aw+cu:bu+av.}

<p>The lines tangent to the general inconic are the sidelines of △<i>ABC</i>, given by the equations <i>x</i> = 0, <i>y</i> = 0, <i>z</i> = 0.
</p>
<h2 id="other-features">Other features</h2>
<h3>Circumconic</h3>
<ul><li>Each noncircular circumconic meets the circumcircle of △<i>ABC</i> in a point other than A, B, C, often called the fourth point of intersection, given by <a href="/facts/Trilinear_coordinates/PLCFluMm">trilinear coordinates</a></li></ul>

(
        c
        x
        −
        a
        z
        )
        (
        a
        y
        −
        b
        x
        )
        :
        (
        a
        y
        −
        b
        x
        )
        (
        b
        z
        −
        c
        y
        )
        :
        (
        b
        z
        −
        c
        y
        )
        (
        c
        x
        −
        a
        z
        )
      
    
    {\displaystyle (cx-az)(ay-bx):(ay-bx)(bz-cy):(bz-cy)(cx-az)}

<ul><li>If 
  
    
      
        P
        =
        p
        :
        q
        :
        r
      
    
    {\displaystyle P=p:q:r}
  
 is a point on the general circumconic, then the line tangent to the conic at P is given by</li></ul>

(
        v
        r
        +
        w
        q
        )
        x
        +
        (
        w
        p
        +
        u
        r
        )
        y
        +
        (
        u
        q
        +
        v
        p
        )
        z
        =
        0.
      
    
    {\displaystyle (vr+wq)x+(wp+ur)y+(uq+vp)z=0.}

<ul><li>The general circumconic reduces to a <a href="/facts/Parabola/Cweo3wgw">parabola</a> if and only if</li></ul>

u
          
            2
          
        
        
          a
          
            2
          
        
        +
        
          v
          
            2
          
        
        
          b
          
            2
          
        
        +
        
          w
          
            2
          
        
        
          c
          
            2
          
        
        −
        2
        v
        w
        b
        c
        −
        2
        w
        u
        c
        a
        −
        2
        u
        v
        a
        b
        =
        0
        ,
      
    
    {\displaystyle u^{2}a^{2}+v^{2}b^{2}+w^{2}c^{2}-2vwbc-2wuca-2uvab=0,}

and to a <a href="/facts/Rectangular_hyperbola/6R6ER0mY">rectangular hyperbola</a> if and only if

u
        cos
        ⁡
        A
        +
        v
        cos
        ⁡
        B
        +
        w
        cos
        ⁡
        C
        =
        0.
      
    
    {\displaystyle u\cos A+v\cos B+w\cos C=0.}

<ul><li>Of all triangles inscribed in a given ellipse, the <a href="/facts/Centroid/H6dDbg0h">centroid</a> of the one with greatest area coincides with the center of the ellipse.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a>: p.147  The given ellipse, going through this triangle's three vertices and centered at the triangle's centroid, is called the triangle's <a href="/facts/Steiner_circumellipse/q1u6Bvbs">Steiner circumellipse</a>.</li></ul>
<h3>Inconic</h3>
<ul><li>The general inconic reduces to a <a href="/facts/Parabola/Cweo3wgw">parabola</a> if and only if</li></ul>

u
        b
        c
        +
        v
        c
        a
        +
        w
        a
        b
        =
        0
        ,
      
    
    {\displaystyle ubc+vca+wab=0,}

in which case it is tangent externally to one of the sides of the triangle and is tangent to the <a href="/facts/Extended_side/wtqKRrBS">extensions of the other two sides</a>.
<ul><li>Suppose that ⁠
  
    
      
        
          p
          
            1
          
        
        :
        
          q
          
            1
          
        
        :
        
          r
          
            1
          
        
      
    
    {\displaystyle p_{1}:q_{1}:r_{1}}
  
⁠ and ⁠
  
    
      
        
          p
          
            2
          
        
        :
        
          q
          
            2
          
        
        :
        
          r
          
            2
          
        
      
    
    {\displaystyle p_{2}:q_{2}:r_{2}}
  
⁠ are distinct points, and let</li></ul>

X
        =
        (
        
          p
          
            1
          
        
        +
        
          p
          
            2
          
        
        t
        )
        :
        (
        
          q
          
            1
          
        
        +
        
          q
          
            2
          
        
        t
        )
        :
        (
        
          r
          
            1
          
        
        +
        
          r
          
            2
          
        
        t
        )
        .
      
    
    {\displaystyle X=(p_{1}+p_{2}t):(q_{1}+q_{2}t):(r_{1}+r_{2}t).}

As the parameter t ranges through the <a href="/facts/Real_numbers/R02gw5Pb">real numbers</a>, the locus of X is a line.  Define

X
          
            2
          
        
        =
        (
        
          p
          
            1
          
        
        +
        
          p
          
            2
          
        
        t
        
          )
          
            2
          
        
        :
        (
        
          q
          
            1
          
        
        +
        
          q
          
            2
          
        
        t
        
          )
          
            2
          
        
        :
        (
        
          r
          
            1
          
        
        +
        
          r
          
            2
          
        
        t
        
          )
          
            2
          
        
        .
      
    
    {\displaystyle X^{2}=(p_{1}+p_{2}t)^{2}:(q_{1}+q_{2}t)^{2}:(r_{1}+r_{2}t)^{2}.}

The locus of <i>X</i>2 is the inconic, necessarily an <a href="/facts/Ellipse/1wUDnGxY">ellipse</a>, given by the equation

L
          
            4
          
        
        
          x
          
            2
          
        
        +
        
          M
          
            4
          
        
        
          y
          
            2
          
        
        +
        
          N
          
            4
          
        
        
          z
          
            2
          
        
        −
        2
        
          M
          
            2
          
        
        
          N
          
            2
          
        
        y
        z
        −
        2
        
          N
          
            2
          
        
        
          L
          
            2
          
        
        z
        x
        −
        2
        
          L
          
            2
          
        
        
          M
          
            2
          
        
        x
        y
        =
        0
        ,
      
    
    {\displaystyle L^{4}x^{2}+M^{4}y^{2}+N^{4}z^{2}-2M^{2}N^{2}yz-2N^{2}L^{2}zx-2L^{2}M^{2}xy=0,}

where

L
              
              
                
                =
                
                  q
                  
                    1
                  
                
                
                  r
                  
                    2
                  
                
                −
                
                  r
                  
                    1
                  
                
                
                  q
                  
                    2
                  
                
                ,
              
            
            
              
                M
              
              
                
                =
                
                  r
                  
                    1
                  
                
                
                  p
                  
                    2
                  
                
                −
                
                  p
                  
                    1
                  
                
                
                  r
                  
                    2
                  
                
                ,
              
            
            
              
                N
              
              
                
                =
                
                  p
                  
                    1
                  
                
                
                  q
                  
                    2
                  
                
                −
                
                  q
                  
                    1
                  
                
                
                  p
                  
                    2
                  
                
                .
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}L&=q_{1}r_{2}-r_{1}q_{2},\\M&=r_{1}p_{2}-p_{1}r_{2},\\N&=p_{1}q_{2}-q_{1}p_{2}.\end{aligned}}}

<ul><li>A point in the interior of a triangle is the center of an inellipse of the triangle if and only if  the point lies in the interior of the triangle whose vertices lie at the midpoints of the original triangle's sides.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a>: p.139  For a given point inside that <a href="/facts/Medial_triangle/QKYBhNQM">medial triangle</a>, the inellipse with its center at that point is unique.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a>: p.142 </li>
<li>The inellipse with the largest area is the <a href="/facts/Steiner_inellipse/IFJ4pLGr">Steiner inellipse</a>, also called the midpoint inellipse, with its center at the triangle's <a href="/facts/Centroid/H6dDbg0h">centroid</a>.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a>: p.145  In general, the ratio of the inellipse's area to the triangle's area, in terms of the unit-sum <a href="/facts/Barycentric_coordinate_system/ymlRaudU">barycentric coordinates</a> (<i>α, β, γ</i>) of the inellipse's center, is<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a>: p.143 </li></ul>

Area of inellipse
            Area of triangle
          
        
        =
        π
        
          
            (
            1
            −
            2
            α
            )
            (
            1
            −
            2
            β
            )
            (
            1
            −
            2
            γ
            )
          
        
        ,
      
    
    {\displaystyle {\frac {\text{Area of inellipse}}{\text{Area of triangle}}}=\pi {\sqrt {(1-2\alpha )(1-2\beta )(1-2\gamma )}},}

which is maximized by the centroid's barycentric coordinates <i>α</i> = <i>β</i> = <i>γ</i> = ⅓.
<ul><li>The lines connecting the tangency points of any inellipse of a triangle with the opposite vertices of the triangle are concurrent.<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a>: p.148 </li></ul>
<h2 id="extension-to-quadrilaterals">Extension to quadrilaterals</h2>
<p>All the centers of inellipses of a given <a href="/facts/Quadrilateral/QCGqrkp2">quadrilateral</a> fall on the line segment connecting the midpoints of the <a href="/facts/Diagonal/39z9sIRb">diagonals</a> of the quadrilateral.<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a>: p.136 
</p>
<h2 id="examples">Examples</h2>
<ul><li>Circumconics
<ul><li><a href="/facts/Circumcircle/RafYT4am">Circumcircle</a>, the unique <a href="/facts/Circle/9fy5ggKn">circle</a> that passes through a triangle's three vertices</li>
<li><a href="/facts/Steiner_circumellipse/q1u6Bvbs">Steiner circumellipse</a>, the unique ellipse that passes through a triangle's three vertices and is centered at the triangle's <a href="/facts/Centroid/H6dDbg0h">centroid</a></li>
<li><a href="/facts/Kiepert_conics/ew4f1pVW">Kiepert hyperbola</a>, the unique conic which passes through a triangle's three vertices, its centroid, and its <a href="/facts/Orthocenter/z9H8fFMz">orthocenter</a></li>
<li><a href="/facts/V%25C3%25A1clav_Je%25C5%2599%25C3%25A1bek/fQDXGh7t">Jeřábek</a> hyperbola, a <a href="/facts/Rectangular_hyperbola/6R6ER0mY">rectangular hyperbola</a> centered on a triangle's <a href="/facts/Nine-point_circle/0UfSxFX9">nine-point circle</a> and passing through the triangle's three vertices as well as its <a href="/facts/Circumcenter/RafYT4am">circumcenter</a>, orthocenter, and various other notable centers</li>
<li><a href="/facts/Feuerbach_hyperbola/Rc1VMl4s">Feuerbach hyperbola</a>, a rectangular hyperbola that passes through a triangle's orthocenter, <a href="/facts/Nagel_point/fGSa5LJd">Nagel point</a>, and various other notable points, and has center on the nine-point circle.</li></ul></li>
<li>Inconics
<ul><li><a href="/facts/Incircle/x9QCn9ML">Incircle</a>, the unique circle that is internally tangent to a triangle's three sides</li>
<li><a href="/facts/Steiner_inellipse/IFJ4pLGr">Steiner inellipse</a>, the unique ellipse that is tangent to a triangle's three sides at their midpoints</li>
<li><a href="/facts/Mandart_inellipse/BjpSTurV">Mandart inellipse</a>, the unique ellipse tangent to a triangle's sides at the contact points of its <a href="/facts/Excircle/x9QCn9ML">excircles</a></li>
<li><a href="/facts/Kiepert_conics/ew4f1pVW">Kiepert parabola</a></li>
<li>Yff parabola</li></ul></li></ul>

<h2 id="external-links">External links</h2>
<ul><li><a href="http://mathworld.wolfram.com/Circumconic.html">Circumconic</a> at MathWorld</li>
<li><a href="http://mathworld.wolfram.com/Inconic.html">Inconic</a> at MathWorld</li></ul>

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

<ol>
<li id="fn:1"><p>Weisstein, Eric W. "Circumconic." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Circumconic.html <a href="http://mathworld.wolfram.com/Circumconic.html" target="_blank">http://mathworld.wolfram.com/Circumconic.html</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Weisstein, Eric W. "Inconic." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Inconic.html <a href="http://mathworld.wolfram.com/Inconic.html" target="_blank">http://mathworld.wolfram.com/Inconic.html</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Chakerian, G. D. "A Distorted View of Geometry." Ch. 7 in Mathematical Plums (R. Honsberger, editor). Washington, DC: Mathematical Association of America, 1979. <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Chakerian, G. D. "A Distorted View of Geometry." Ch. 7 in Mathematical Plums (R. Honsberger, editor). Washington, DC: Mathematical Association of America, 1979. <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Chakerian, G. D. "A Distorted View of Geometry." Ch. 7 in Mathematical Plums (R. Honsberger, editor). Washington, DC: Mathematical Association of America, 1979. <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>Chakerian, G. D. "A Distorted View of Geometry." Ch. 7 in Mathematical Plums (R. Honsberger, editor). Washington, DC: Mathematical Association of America, 1979. <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
<li id="fn:7"><p>Chakerian, G. D. "A Distorted View of Geometry." Ch. 7 in Mathematical Plums (R. Honsberger, editor). Washington, DC: Mathematical Association of America, 1979. <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li>
<li id="fn:8"><p>Chakerian, G. D. "A Distorted View of Geometry." Ch. 7 in Mathematical Plums (R. Honsberger, editor). Washington, DC: Mathematical Association of America, 1979. <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li>
<li id="fn:9"><p>Chakerian, G. D. "A Distorted View of Geometry." Ch. 7 in Mathematical Plums (R. Honsberger, editor). Washington, DC: Mathematical Association of America, 1979. <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li>
</ol>

Circumconic and inconic open-in-new

Circumconic and inconic