Difference between revisions of "Brocard point"
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<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> | <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> | ||
− | 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 | + | 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 |
<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> | <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> |
Revision as of 19:21, 6 November 2016
The first (or positive) Brocard point of a plane triangle with vertices
,
,
is the interior point
of
for which the three angles
,
,
are equal. Their common value
is the Brocard angle of
.
The second (or negative) Brocard point of is the interior point
for which
. Their common value is again
. The Brocard angle satisfies
. The two Brocard points are isogonal conjugates (cf. Isogonal); they coincide if
is equilateral, in which case
.
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 and
, 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 be an arbitrary plane triangle with vertices
,
,
and angles
,
,
. If
denotes the circle that is tangent to the line
at
and passes through the vertices
and
, then
also passes through
. Similarly for the circles
and
. So the three circles
,
,
intersect in the first Brocard point
. Analogously, the circle
that passes through
and
and is tangent to the line
at
, meets the circles
and
in the second Brocard point
. Further, the circumcentre
of
and the two Brocard points are vertices of a isosceles triangle for which
. The lengths of the sides of this triangle can be expressed in terms of the radius
of the circumcircle of
, and the Brocard angle
:
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The Brocard circle is the circle passing through the two Brocard points and . The Lemoine point
of
, named after E. Lemoine, is a distinguished point of this circle, and the length of the line segment
![]() |
gives the diameter of the Brocard circle.
The Brocard angle is related to the three angles
,
,
by the following trigonometric identities:
![]() |
![]() |
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,
![]() |
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 |
Brocard point. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Brocard_point&oldid=39659