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The first (or positive) Brocard point of a plane triangle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130250/b1302501.png" /> with vertices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130250/b1302502.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130250/b1302503.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130250/b1302504.png" /> is the interior point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130250/b1302505.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130250/b1302506.png" /> for which the three angles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130250/b1302507.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130250/b1302508.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130250/b1302509.png" /> are equal. Their common value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130250/b13025010.png" /> is the Brocard angle of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130250/b13025011.png" />.
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The second (or negative) Brocard point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130250/b13025012.png" /> is the interior point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130250/b13025013.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130250/b13025014.png" />. Their common value is again <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130250/b13025015.png" />. The Brocard angle satisfies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130250/b13025016.png" />. The two Brocard points are isogonal conjugates (cf. [[Isogonal|Isogonal]]); they coincide if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130250/b13025017.png" /> is equilateral, in which case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130250/b13025018.png" />.
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The first (or positive) Brocard point of a plane triangle $( T )$ with vertices $A$, $B$, $C$ is the interior point $\Omega$ of $( T )$ for which the three angles $\angle \Omega A B$, $\angle \Omega B C$, $\angle \Omega C A$ are equal. Their common value $\omega$ is the Brocard angle of $( T )$.
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The second (or negative) Brocard point of $( T )$ is the interior point $\Omega ^ { \prime }$ for which $\angle \Omega ^ { \prime } B A = \angle \Omega ^ { \prime } C B = \angle \Omega ^ { \prime } A C$. Their common value is again $\omega$. The Brocard angle satisfies $0 &lt; \omega \leq \pi / 6$. The two Brocard points are isogonal conjugates (cf. [[Isogonal|Isogonal]]); they coincide if $( T )$ is equilateral, in which case $\omega = \pi / 6$.
  
 
The Brocard configuration (for an extensive account see [[#References|[a6]]]), named after H. Brocard who first published about it around 1875, belongs to triangle geometry, a subbranch of Euclidean geometry that thrived in the last quarter of the nineteenth century to fade away again in the first quarter of the twentieth century. A brief historical account is given in [[#References|[a5]]].
 
The Brocard configuration (for an extensive account see [[#References|[a6]]]), named after H. Brocard who first published about it around 1875, belongs to triangle geometry, a subbranch of Euclidean geometry that thrived in the last quarter of the nineteenth century to fade away again in the first quarter of the twentieth century. A brief historical account is given in [[#References|[a5]]].
  
Although his name is generally associated with the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130250/b13025019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130250/b13025020.png" />, Brocard was not the first person to investigate their properties; in 1816, long before Brocard wrote about them, they were mentioned by A.L. Crelle in [[#References|[a4]]] (see also [[#References|[a8]]] and [[#References|[a11]]]). Information on Brocard's life can be found in [[#References|[a7]]].
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Although his name is generally associated with the points $\Omega$ and $\Omega ^ { \prime }$, Brocard was not the first person to investigate their properties; in 1816, long before Brocard wrote about them, they were mentioned by A.L. Crelle in [[#References|[a4]]] (see also [[#References|[a8]]] and [[#References|[a11]]]). Information on Brocard's life can be found in [[#References|[a7]]].
  
 
The Brocard points and Brocard angle have many remarkable properties. Some characteristics of the Brocard configuration are given below.
 
The Brocard points and Brocard angle have many remarkable properties. Some characteristics of the Brocard configuration are given below.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130250/b13025021.png" /> be an arbitrary plane triangle with vertices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130250/b13025022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130250/b13025023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130250/b13025024.png" /> and angles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130250/b13025025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130250/b13025026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130250/b13025027.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130250/b13025028.png" /> denotes the circle that is tangent to the line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130250/b13025029.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130250/b13025030.png" /> and passes through the vertices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130250/b13025031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130250/b13025032.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130250/b13025033.png" /> also passes through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130250/b13025034.png" />. Similarly for the circles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130250/b13025035.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130250/b13025036.png" />. So the three circles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130250/b13025037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130250/b13025038.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130250/b13025039.png" /> intersect in the first Brocard point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130250/b13025040.png" />. Analogously, the circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130250/b13025041.png" /> that passes through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130250/b13025042.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130250/b13025043.png" /> and is tangent to the line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130250/b13025044.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130250/b13025045.png" />, meets the circles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130250/b13025046.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130250/b13025047.png" /> in the second Brocard point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130250/b13025048.png" />. Further, the [[circumcentre]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130250/b13025049.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130250/b13025050.png" /> and the two Brocard points are vertices of a isosceles triangle for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130250/b13025051.png" />. The lengths of the sides of this triangle can be expressed in terms of the radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130250/b13025052.png" /> of the circumcircle of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130250/b13025053.png" />, and the Brocard angle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130250/b13025054.png" />:
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Let $( T )$ be an arbitrary plane triangle with vertices $A$, $B$, $C$ and angles $\alpha = \angle B A C$, $\beta = \angle C B A$, $\gamma = \angle A C B$. If $C _ { B C }$ denotes the circle that is tangent to the line $A C$ at $C$ and passes through the vertices $B$ and $C$, then $C _ { B C }$ also passes through $\Omega$. Similarly for the circles $C _ { C A }$ and $C _ { A B }$. So the three circles $C _ { B C }$, $C _ { C A }$, $C _ { A B }$ intersect in the first Brocard point $\Omega$. Analogously, the circle $C ^ { \prime_{ BC}}$ that passes through $B$ and $C$ and is tangent to the line $A B$ at $B$, meets the circles $C ^ { \prime CA }$ and $C ^ { \prime _{ AB}}$ in the second Brocard point $\Omega ^ { \prime }$. Further, the [[circumcentre]] $O$ of $( T )$ and the two Brocard points are vertices of a isosceles triangle for which $\angle \Omega O \Omega ^ { \prime } = 2 \omega$. The lengths of the sides of this triangle can be expressed in terms of the radius $R$ of the circumcircle of $( T )$, and the Brocard angle $\omega$:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130250/b13025055.png" /></td> </tr></table>
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\begin{equation*} \frac { \overline { \Omega  \Omega ^ { \prime } }  } { 2 \operatorname { sin } \omega } = \overline { O \Omega } = \overline { O \Omega ^ { \prime } } = R \sqrt { 1 - 4 \operatorname { sin } ^ { 2 } \omega }. \end{equation*}
  
The Brocard circle is the circle passing through the two Brocard points and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130250/b13025056.png" />. The Lemoine point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130250/b13025057.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130250/b13025058.png" />, named after E. Lemoine, is a distinguished point of this circle, and the length of the line segment
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The Brocard circle is the circle passing through the two Brocard points and $O$. The [[Lemoine point]] $K$ of $( T )$, named after E. Lemoine, is a distinguished point of this circle, and the length of the line segment
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130250/b13025059.png" /></td> </tr></table>
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\begin{equation*} \overline { O K } = \frac { \overline { O \Omega } } { \operatorname { cos } \omega } \end{equation*}
  
 
gives the diameter of the Brocard circle.
 
gives the diameter of the Brocard circle.
  
The Brocard angle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130250/b13025060.png" /> is related to the three angles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130250/b13025061.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130250/b13025062.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130250/b13025063.png" /> by the following trigonometric identities:
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The Brocard angle $\omega$ is related to the three angles $\alpha$, $\beta$, $\gamma$ by the following trigonometric identities:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130250/b13025064.png" /></td> </tr></table>
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\begin{equation*} \operatorname { cot } \omega = \operatorname { cot } \alpha + \operatorname { cot } \beta + \operatorname { cot } \gamma, \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130250/b13025065.png" /></td> </tr></table>
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\begin{equation*} \frac { 1 } { \operatorname { sin } ^ { 2 } \omega } = \frac { 1 } { \operatorname { sin } ^ { 2 } \alpha } + \frac { 1 } { \operatorname { sin } ^ { 2 } \beta } + \frac { 1 } { \operatorname { sin } ^ { 2 } \gamma }. \end{equation*}
  
 
Due to a remarkable conjecture by P. Yff in 1963 (see [[#References|[a14]]]), modest interest in the Brocard configuration arose again during the 1960s, 1970s and 1980s. This conjecture, known as Yff's inequality,
 
Due to a remarkable conjecture by P. Yff in 1963 (see [[#References|[a14]]]), modest interest in the Brocard configuration arose again during the 1960s, 1970s and 1980s. This conjecture, known as Yff's inequality,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130250/b13025066.png" /></td> </tr></table>
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\begin{equation*} 8 \omega ^ { 3 } \leq \alpha \, \beta \, \gamma , \end{equation*}
  
 
is unusual in the sense that it contains the angles proper instead of their trigonometric function values (as could be expected). A proof for this conjecture was found by F. Abi-Khuzam in 1974 (see [[#References|[a2]]]). In [[#References|[a12]]] and [[#References|[a1]]] a few inequalities of similar type were proposed and subsequently proven.
 
is unusual in the sense that it contains the angles proper instead of their trigonometric function values (as could be expected). A proof for this conjecture was found by F. Abi-Khuzam in 1974 (see [[#References|[a2]]]). In [[#References|[a12]]] and [[#References|[a1]]] a few inequalities of similar type were proposed and subsequently proven.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  F.F. Abi–Khuzam,  A.B. Boghossian,  "Some recent geometric inequalities"  ''Amer. Math. Monthly'' , '''96'''  (1989)  pp. 576–589</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  F. Abi–Khuzam,  "Proof of Yff's conjecture on the Brocard angle of a triangle"  ''Elem. Math.'' , '''29'''  (1974)  pp. 141–142</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J. Casey,  "Géometrie elementaire récente" , Gauthier-Villars  (1890)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A.L. Crelle,  "Über einige Eigenschaften des ebenen geradlinigen Dreiecks rücksichtlich dreier durch die Winkelspitzen gezogenen geraden Linien" , Berlin  (1816)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  Ph.J. Davis,  "The rise, fall, and possible transfiguration of triangle geometry: A mini-history"  ''Amer. Math. Monthly'' , '''102'''  (1995)  pp. 204–214</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  A. Emmerich,  "Die Brocardschen Gebilde und ihre Beziehungen zu den verwandten merkwürdigen Punkten und Kreisen des Dreiecks" , G. Reimer  (1891)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  L. Guggenbuhl,  "Henri Brocard and the geometry of the triangle"  ''Math. Gazette'' , '''80'''  (1996)  pp. 492–500</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  R. Honsberger,  "The Brocard angle" , ''Episodes in Nineteenth and Twentieth Century Euclidean Geometry'' , Math. Assoc. America  (1995)  pp. 101–106</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  R.A. Johnson,  "Modern geometry: an elementary treatise on the geometry of the triangle and the circle" , Houghton–Mifflin  (1929)  (Reprinted as: Advanced Euclidean Geometry, Dover,1960)</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  C. Kimberling,  "Central points and central lines in the plane of a triangle"  ''Math. Mag.'' , '''67'''  (1994)  pp. 163–187</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  D. Mitrinović,  J.E. Pečarić,  V. Volenec,  "Recent advances in geometric inequalities" , Kluwer Acad. Publ.  (1989)</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  R.J. Stroeker,  H.J.T. Hoogland,  "Brocardian geometry revisited or some remarkable inequalities"  ''Nieuw Arch. Wisk. 4th Ser.'' , '''2'''  (1984)  pp. 281–310</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  R.J. Stroeker,  "Brocard points, circulant matrices, and Descartes' folium"  ''Math. Mag.'' , '''61'''  (1988)  pp. 172–187</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top">  P. Yff,  "An analogue of the Brocard points"  ''Amer. Math. Monthly'' , '''70'''  (1963)  pp. 495–501</TD></TR></table>
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<table>
 +
<tr><td valign="top">[a1]</td> <td valign="top">  F.F. Abi–Khuzam,  A.B. Boghossian,  "Some recent geometric inequalities"  ''Amer. Math. Monthly'' , '''96'''  (1989)  pp. 576–589</td></tr>
 +
<tr><td valign="top">[a2]</td> <td valign="top">  F. Abi–Khuzam,  "Proof of Yff's conjecture on the Brocard angle of a triangle"  ''Elem. Math.'' , '''29'''  (1974)  pp. 141–142</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  J. Casey,  "Géométrie élémentaire récente" , Gauthier-Villars  (1890)</td></tr>
 +
<tr><td valign="top">[a4]</td> <td valign="top">  A.L. Crelle,  "Über einige Eigenschaften des ebenen geradlinigen Dreiecks rücksichtlich dreier durch die Winkelspitzen gezogenen geraden Linien" , Berlin  (1816)</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  Ph.J. Davis,  "The rise, fall, and possible transfiguration of triangle geometry: A mini-history"  ''Amer. Math. Monthly'' , '''102'''  (1995)  pp. 204–214</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  A. Emmerich,  "Die Brocardschen Gebilde und ihre Beziehungen zu den verwandten merkwürdigen Punkten und Kreisen des Dreiecks" , G. Reimer  (1891)</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  L. Guggenbuhl,  "Henri Brocard and the geometry of the triangle"  ''Math. Gazette'' , '''80'''  (1996)  pp. 492–500</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  R. Honsberger,  "The Brocard angle" , ''Episodes in Nineteenth and Twentieth Century Euclidean Geometry'' , Math. Assoc. America  (1995)  pp. 101–106</td></tr><tr><td valign="top">[a9]</td> <td valign="top">  R.A. Johnson,  "Modern geometry: an elementary treatise on the geometry of the triangle and the circle" , Houghton–Mifflin  (1929)  (Reprinted as: Advanced Euclidean Geometry, Dover,1960)</td></tr><tr><td valign="top">[a10]</td> <td valign="top">  C. Kimberling,  "Central points and central lines in the plane of a triangle"  ''Math. Mag.'' , '''67'''  (1994)  pp. 163–187</td></tr><tr><td valign="top">[a11]</td> <td valign="top">  D. Mitrinović,  J.E. Pečarić,  V. Volenec,  "Recent advances in geometric inequalities" , Kluwer Acad. Publ.  (1989)</td></tr><tr><td valign="top">[a12]</td> <td valign="top">  R.J. Stroeker,  H.J.T. Hoogland,  "Brocardian geometry revisited or some remarkable inequalities"  ''Nieuw Arch. Wisk. 4th Ser.'' , '''2'''  (1984)  pp. 281–310</td></tr><tr><td valign="top">[a13]</td> <td valign="top">  R.J. Stroeker,  "Brocard points, circulant matrices, and Descartes' folium"  ''Math. Mag.'' , '''61'''  (1988)  pp. 172–187</td></tr><tr><td valign="top">[a14]</td> <td valign="top">  P. Yff,  "An analogue of the Brocard points"  ''Amer. Math. Monthly'' , '''70'''  (1963)  pp. 495–501</td></tr>
 +
</table>

Latest revision as of 05:53, 17 July 2024

The first (or positive) Brocard point of a plane triangle $( T )$ with vertices $A$, $B$, $C$ is the interior point $\Omega$ of $( T )$ for which the three angles $\angle \Omega A B$, $\angle \Omega B C$, $\angle \Omega C A$ are equal. Their common value $\omega$ is the Brocard angle of $( T )$.

The second (or negative) Brocard point of $( T )$ is the interior point $\Omega ^ { \prime }$ for which $\angle \Omega ^ { \prime } B A = \angle \Omega ^ { \prime } C B = \angle \Omega ^ { \prime } A C$. Their common value is again $\omega$. The Brocard angle satisfies $0 < \omega \leq \pi / 6$. The two Brocard points are isogonal conjugates (cf. Isogonal); they coincide if $( T )$ is equilateral, in which case $\omega = \pi / 6$.

The Brocard configuration (for an extensive account see [a6]), named after H. Brocard who first published about it around 1875, belongs to triangle geometry, a subbranch of Euclidean geometry that thrived in the last quarter of the nineteenth century to fade away again in the first quarter of the twentieth century. A brief historical account is given in [a5].

Although his name is generally associated with the points $\Omega$ and $\Omega ^ { \prime }$, Brocard was not the first person to investigate their properties; in 1816, long before Brocard wrote about them, they were mentioned by A.L. Crelle in [a4] (see also [a8] and [a11]). Information on Brocard's life can be found in [a7].

The Brocard points and Brocard angle have many remarkable properties. Some characteristics of the Brocard configuration are given below.

Let $( T )$ be an arbitrary plane triangle with vertices $A$, $B$, $C$ and angles $\alpha = \angle B A C$, $\beta = \angle C B A$, $\gamma = \angle A C B$. If $C _ { B C }$ denotes the circle that is tangent to the line $A C$ at $C$ and passes through the vertices $B$ and $C$, then $C _ { B C }$ also passes through $\Omega$. Similarly for the circles $C _ { C A }$ and $C _ { A B }$. So the three circles $C _ { B C }$, $C _ { C A }$, $C _ { A B }$ intersect in the first Brocard point $\Omega$. Analogously, the circle $C ^ { \prime_{ BC}}$ that passes through $B$ and $C$ and is tangent to the line $A B$ at $B$, meets the circles $C ^ { \prime CA }$ and $C ^ { \prime _{ AB}}$ in the second Brocard point $\Omega ^ { \prime }$. Further, the circumcentre $O$ of $( T )$ and the two Brocard points are vertices of a isosceles triangle for which $\angle \Omega O \Omega ^ { \prime } = 2 \omega$. The lengths of the sides of this triangle can be expressed in terms of the radius $R$ of the circumcircle of $( T )$, and the Brocard angle $\omega$:

\begin{equation*} \frac { \overline { \Omega \Omega ^ { \prime } } } { 2 \operatorname { sin } \omega } = \overline { O \Omega } = \overline { O \Omega ^ { \prime } } = R \sqrt { 1 - 4 \operatorname { sin } ^ { 2 } \omega }. \end{equation*}

The Brocard circle is the circle passing through the two Brocard points and $O$. The Lemoine point $K$ of $( T )$, named after E. Lemoine, is a distinguished point of this circle, and the length of the line segment

\begin{equation*} \overline { O K } = \frac { \overline { O \Omega } } { \operatorname { cos } \omega } \end{equation*}

gives the diameter of the Brocard circle.

The Brocard angle $\omega$ is related to the three angles $\alpha$, $\beta$, $\gamma$ by the following trigonometric identities:

\begin{equation*} \operatorname { cot } \omega = \operatorname { cot } \alpha + \operatorname { cot } \beta + \operatorname { cot } \gamma, \end{equation*}

\begin{equation*} \frac { 1 } { \operatorname { sin } ^ { 2 } \omega } = \frac { 1 } { \operatorname { sin } ^ { 2 } \alpha } + \frac { 1 } { \operatorname { sin } ^ { 2 } \beta } + \frac { 1 } { \operatorname { sin } ^ { 2 } \gamma }. \end{equation*}

Due to a remarkable conjecture by P. Yff in 1963 (see [a14]), modest interest in the Brocard configuration arose again during the 1960s, 1970s and 1980s. This conjecture, known as Yff's inequality,

\begin{equation*} 8 \omega ^ { 3 } \leq \alpha \, \beta \, \gamma , \end{equation*}

is unusual in the sense that it contains the angles proper instead of their trigonometric function values (as could be expected). A proof for this conjecture was found by F. Abi-Khuzam in 1974 (see [a2]). In [a12] and [a1] a few inequalities of similar type were proposed and subsequently proven.

References

[a1] F.F. Abi–Khuzam, A.B. Boghossian, "Some recent geometric inequalities" Amer. Math. Monthly , 96 (1989) pp. 576–589
[a2] F. Abi–Khuzam, "Proof of Yff's conjecture on the Brocard angle of a triangle" Elem. Math. , 29 (1974) pp. 141–142
[a3] J. Casey, "Géométrie élémentaire récente" , Gauthier-Villars (1890)
[a4] A.L. Crelle, "Über einige Eigenschaften des ebenen geradlinigen Dreiecks rücksichtlich dreier durch die Winkelspitzen gezogenen geraden Linien" , Berlin (1816)
[a5] Ph.J. Davis, "The rise, fall, and possible transfiguration of triangle geometry: A mini-history" Amer. Math. Monthly , 102 (1995) pp. 204–214
[a6] A. Emmerich, "Die Brocardschen Gebilde und ihre Beziehungen zu den verwandten merkwürdigen Punkten und Kreisen des Dreiecks" , G. Reimer (1891)
[a7] L. Guggenbuhl, "Henri Brocard and the geometry of the triangle" Math. Gazette , 80 (1996) pp. 492–500
[a8] R. Honsberger, "The Brocard angle" , Episodes in Nineteenth and Twentieth Century Euclidean Geometry , Math. Assoc. America (1995) pp. 101–106
[a9] R.A. Johnson, "Modern geometry: an elementary treatise on the geometry of the triangle and the circle" , Houghton–Mifflin (1929) (Reprinted as: Advanced Euclidean Geometry, Dover,1960)
[a10] C. Kimberling, "Central points and central lines in the plane of a triangle" Math. Mag. , 67 (1994) pp. 163–187
[a11] D. Mitrinović, J.E. Pečarić, V. Volenec, "Recent advances in geometric inequalities" , Kluwer Acad. Publ. (1989)
[a12] R.J. Stroeker, H.J.T. Hoogland, "Brocardian geometry revisited or some remarkable inequalities" Nieuw Arch. Wisk. 4th Ser. , 2 (1984) pp. 281–310
[a13] R.J. Stroeker, "Brocard points, circulant matrices, and Descartes' folium" Math. Mag. , 61 (1988) pp. 172–187
[a14] P. Yff, "An analogue of the Brocard points" Amer. Math. Monthly , 70 (1963) pp. 495–501
How to Cite This Entry:
Brocard point. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Brocard_point&oldid=39659
This article was adapted from an original article by R.J. Stroeker (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article