Brocard Triangles
Let the point of intersection of 
and 
be 
, where 
and 
are the BrocardPoints, and similarly define 
and 
. 
is the first Brocard triangle, and is inversely similar to
. It is inscribed in the Brocard Circle drawn with 
as the Diameter. The triangles 
, 
,and 
are Isosceles Triangles with base angles 
, where is theBrocard Angle. The sum of the areas of the Isosceles Triangles is 
, the Area of Triangle. The first Brocard triangle is in perspective with the given Triangle, with 
, 
, and 
Concurrent. The Median Point of the first Brocard triangle is the Median Point 
of the originaltriangle. The Brocard triangles are in perspective at .
Let 
, 
, and 
and 
, 
, and 
be the Circles intersecting in the BrocardPoints and , respectively. Let the two circles and tangent at 
to 
and 
,and passing respectively through 
and 
, meet again at 
. The triangle 
is the second Brocard triangle.Each Vertex of the second Brocard triangle lies on the second Brocard Circle.
The two Brocard triangles are in perspective at .
References
Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 277-281, 1929.
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