In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
, Brocard points are special points within a
triangle
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, any three points, when non- colline ...
. They are named after
Henri Brocard
Pierre René Jean Baptiste Henri Brocard (12 May 1845 – 16 January 1922) was a French meteorologist and mathematician, in particular a geometer. His best-known achievement is the invention and discovery of the properties of the Brocard point ...
(1845–1922), a French mathematician.
Definition
In a triangle ''ABC'' with sides ''a'', ''b'', and ''c'', where the vertices are labeled ''A'', ''B'' and ''C'' in counterclockwise order, there is exactly one point ''P'' such that the line segments ''AP'', ''BP'', and ''CP'' form the same angle, ω, with the respective sides ''c'', ''a'', and ''b'', namely that
:
Point ''P'' is called the first Brocard point of the triangle ''ABC'', and the angle ''ω'' is called the Brocard angle of the triangle. This angle has the property that
:
where
are the vertex angles
respectively.
There is also a second Brocard point, Q, in triangle ''ABC'' such that line segments ''AQ'', ''BQ'', and ''CQ'' form equal angles with sides ''b'', ''c'', and ''a'' respectively. In other words, the equations
apply. Remarkably, this second Brocard point has the same Brocard angle as the first Brocard point. In other words, angle
is the same as
The two Brocard points are closely related to one another; In fact, the difference between the first and the second depends on the order in which the angles of triangle ''ABC'' are taken. So for example, the first Brocard point of triangle ''ABC'' is the same as the second Brocard point of triangle ''ACB''.
The two Brocard points of a triangle ''ABC'' are
isogonal conjugate __notoc__
In geometry, the isogonal conjugate of a point with respect to a triangle is constructed by reflecting the lines about the angle bisectors of respectively. These three reflected lines concur at the isogonal conjugate of . (Th ...
s of each other.
Construction
The most elegant construction of the Brocard points goes as follows. In the following example the first Brocard point is presented, but the construction for the second Brocard point is very similar.
As in the diagram above, form a circle through points A and B, tangent to edge BC of the triangle (the center of this circle is at the point where the perpendicular bisector of AB meets the line through point B that is perpendicular to BC). Symmetrically, form a circle through points B and C, tangent to edge AC, and a circle through points A and C, tangent to edge AB. These three circles have a common point, the first Brocard point of triangle ''ABC''. See also
Tangent lines to circles
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...
.
The three circles just constructed are also designated as epicycles of triangle ''ABC''. The second Brocard point is constructed in similar fashion.
Trilinears and barycentrics of the first two Brocard points
Homogeneous
trilinear coordinates
In geometry, the trilinear coordinates of a point relative to a given triangle describe the relative directed distances from the three sidelines of the triangle. Trilinear coordinates are an example of homogeneous coordinates. The ratio is ...
for the first and second Brocard points are
and
respectively. Thus their
barycentric coordinates
In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related ...
are respectively
[Scott, J. A. "Some examples of the use of areal coordinates in triangle geometry", '']Mathematical Gazette
''The Mathematical Gazette'' is an academic journal of mathematics education, published three times yearly, that publishes "articles about the teaching and learning of mathematics with a focus on the 15–20 age range and expositions of attractive ...
'' 83, November 1999, 472–477. and
The segment between the first two Brocard points
The Brocard points are an example of a bicentric pair of points, but they are not
triangle center
In geometry, a triangle center (or triangle centre) is a point in the plane that is in some sense a center of a triangle akin to the centers of squares and circles, that is, a point that is in the middle of the figure by some measure. For exampl ...
s because neither Brocard point is invariant under
similarity transformations: reflecting a scalene triangle, a special case of a similarity, turns one Brocard point into the other. However, the
unordered pair In mathematics, an unordered pair or pair set is a set of the form , i.e. a set having two elements ''a'' and ''b'' with no particular relation between them, where = . In contrast, an ordered pair (''a'', ''b'') has ''a'' as its first ...
formed by both points is invariant under similarities. The midpoint of the two Brocard points, called the Brocard midpoint, has
trilinear coordinates
In geometry, the trilinear coordinates of a point relative to a given triangle describe the relative directed distances from the three sidelines of the triangle. Trilinear coordinates are an example of homogeneous coordinates. The ratio is ...
:
and is a triangle center; it is center X(39) in the
Encyclopedia of Triangle Centers
The Encyclopedia of Triangle Centers (ETC) is an online list of thousands of points or " centers" associated with the geometry of a triangle. It is maintained by Clark Kimberling, Professor of Mathematics at the University of Evansville.
, the ...
. The third Brocard point, given in trilinear coordinates as
:
[Entry X(76) in th]
Encyclopedia of Triangle Centers
is the Brocard midpoint of the
anticomplementary triangle and is also the
isotomic conjugate
In geometry, the isotomic conjugate of a point with respect to a triangle is another point, defined in a specific way from and : If the base points of the lines on the sides opposite are reflected about the midpoints of their respective sid ...
of the
symmedian point
In geometry, symmedians are three particular lines associated with every triangle. They are constructed by taking a median of the triangle (a line connecting a vertex with the midpoint of the opposite side), and reflecting the line over the co ...
. It is center X(76) in the
Encyclopedia of Triangle Centers
The Encyclopedia of Triangle Centers (ETC) is an online list of thousands of points or " centers" associated with the geometry of a triangle. It is maintained by Clark Kimberling, Professor of Mathematics at the University of Evansville.
, the ...
.
The distance between the first two Brocard points ''P'' and ''Q'' is always less than or equal to half the radius ''R'' of the triangle's
circumcircle
In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius.
Not every polyg ...
:
[Weisstein, Eric W. "Brocard Points." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/BrocardPoints.html]
:
The segment between the first two Brocard points is perpendicularly bisected at the Brocard midpoint by the line connecting the triangle's circumcenter
In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius.
Not every polyg ...
and its Lemoine point
In geometry, the Lemoine point, Grebe point or symmedian point is the intersection of the three symmedians ( medians reflected at the associated angle bisectors) of a triangle.
Ross Honsberger called its existence "one of the crown jewels of ...
. Moreover, the circumcenter, the Lemoine point, and the first two Brocard points are concyclic
In geometry, a set of points are said to be concyclic (or cocyclic) if they lie on a common circle. All concyclic points are at the same distance from the center of the circle. Three points in the plane that do not all fall on a straight line ...
—they all fall on the same circle, of which the segment connecting the circumcenter and the Lemoine point is a diameter
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid fo ...
.[
]
Distance from circumcenter
The Brocard points ''P'' and ''Q'' are equidistant from the triangle's circumcenter
In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius.
Not every polyg ...
''O'':[
:
]
Similarities and congruences
The pedal triangle
In geometry, a pedal triangle is obtained by projecting a point onto the sides of a triangle.
More specifically, consider a triangle ''ABC'', and a point ''P'' that is not one of the vertices ''A, B, C''. Drop perpendiculars from ''P'' to the th ...
s of the first and second Brocard points are congruent
Congruence may refer to:
Mathematics
* Congruence (geometry), being the same size and shape
* Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure
* In mod ...
to each other and similar to the original triangle.[
If the lines ''AP'', ''BP'', and ''CP'', each through one of a triangle's vertices and its first Brocard point, intersect the triangle's ]circumcircle
In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius.
Not every polyg ...
at points ''L'', ''M'', and ''N'', then the triangle ''LMN'' is congruent with the original triangle ''ABC''. The same is true if the first Brocard point ''P'' is replaced by the second Brocard point ''Q''.[
]
Notes
References
*.
*.
{{refend
External links
Third Brocard Point
at MathWorld
Bicentric Pairs of Points and Related Triangle Centers
at MathWorld
Points defined for a triangle