TOPICS

# Second Brocard Triangle

Let , , and be the circles through the vertices and , and , and and , respectively, which intersect in the first Brocard point . Similarly, define , , and with respect to the second Brocard point . Let the two circles and tangent at to and , and passing respectively through and , meet again at , and similarly for and . Then the triangle is called the second Brocard triangle.

The second Brocard triangle is also the triangle obtained as the intersections of the lines , , and with the Brocard circle, where is the symmedian point. Let , , and be the intersections of the lines , , and with the circumcircle of . Then , , and are the midpoints of , , and , respectively (Lachlan 1893).

The second Brocard triangle has trilinear vertex matrix

 (1)

It has area

 (2)

where is the area of the reference triangle, and side lengths

 (3) (4) (5)

where , , and are the side lengths of the reference triangle.

The following table gives the centers of the second Brocard triangle in terms of the centers of the reference triangle that correspond to Kimberling centers .

 center of second Brocard triangle center of reference triangle circumcenter midpoint of Brocard diameter symmedian point harmonic of first isodynamic point first isodynamic point second isodynamic point second isodynamic point Schoute center Schoute center isogonal conjugate of isogonal conjugate of isogonal conjugate of isogonal conjugate of first Brocard-axis intercept of circumcircle symmedian point second Brocard-axis intercept of circumcircle circumcenter second Brocard-axis-Moses-circle intersection Brocard midpoint inverse-in-circumcircle of inverse-in-circumcircle of inverse-in-circumcircle of inverse-in-circumcircle of

The first and second Brocard triangles are in perspective with perspector at the triangle centroid of .

Brocard Triangles, D-Triangle, First Brocard Triangle, Third Brocard Triangle

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## References

Gibert, B. "Brocard Triangles." http://perso.wanadoo.fr/bernard.gibert/gloss/brocardtriangles.html.Honsberger, R. "The Brocard Triangles." §10.4 in Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 110-118, 1995.Lachlan, R. An Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 78-81, 1893.

## Referenced on Wolfram|Alpha

Second Brocard Triangle

## Cite this as:

Weisstein, Eric W. "Second Brocard Triangle." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SecondBrocardTriangle.html