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# First Brocard Triangle

Given triangle , let the point of intersection of and be , where and are the Brocard points, and similarly define and . Then is called the first Brocard triangle, and is inversely similar to (Honsberger 1995, p. 112). It is inscribed in the Brocard circle.

 (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 first Brocard triangle in terms of the centers of the reference triangle for Kimberling centers with .

 center of first Brocard triangle center of reference triangle triangle centroid triangle centroid circumcenter midpoint of Brocard diameter orthocenter reflection of in Exeter point inverse of in the Brocard circle far-out point focus of Kiepert parabola Euler infinity point direction of vector , where symmedian point of the anticomplementary triangle external similitude center of Moses circle and (, ) Tarry point circumcenter Steiner point symmedian point

The triangles , , and are isosceles triangles with base angles , where is the Brocard angle. The sum of the areas of the isosceles triangles is , the area of triangle .

The first Brocard triangle is in perspective with with perspector at the third Brocard point of .

Let perpendiculars be drawn from the midpoints , , and of each side of the first Brocard triangle to the opposite sides of the triangle . Then the extensions of these lines concur in the nine-point center of (Honsberger 1995, pp. 116-118).

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

The triangle centroid of the first Brocard triangle is also the triangle centroid of the original triangle (Honsberger 1995, pp. 112-116).

D-Triangle, Brocard Triangles, Second Brocard Triangle, Third Brocard Triangle

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## References

Gallatly, W. The Modern Geometry of the Triangle, 2nd ed. London: Hodgson, p. 100, 1913.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.

## Referenced on Wolfram|Alpha

First Brocard Triangle

## Cite this as:

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