Brocard point

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, "Géometrie elementaire 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