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The -triangle, also known as the fourth Brocard triangle, is the triangle whose vertex is the second intersection of the circles and used in the construction of the Brocard points, which are tangent to at and , respectively (Gibert).

It has trilinear vertex matrix

The vertices of the -triangle are the isogonal conjugates of the second Brocard triangle (Johnson 1929, p. 285). For a nonequilateral reference triangle, is inversely similar to its circumcevian triangle with respect to the symmedian point (Kimberling). The vertices lie on the respective medians of the reference triangle. The circumcircle of the -triangle is the orthocentroidal circle, which has diameter , where is the triangle centroid and is the orthocenter.

The vertices satisfy

(correcting Johnson 1929, p. 285).

The vertices of the D-triangle lie on the respective Apollonius circles.

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

See also

Apollonius Circle, Brocard Points, Circumcevian Triangle, Second Brocard Triangle, Symmedian Point, Third Brocard Triangle

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References

Gibert, B. "Brocard Triangles." http://bernard-gibert.fr/gloss/brocardtriangles.html.Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 284-285, 296 and 307, 1929.Kimberling, C. "Index of Triangles Referenced in ETC." https://faculty.evansville.edu/ck6/encyclopedia/IndexOfTrianglesReferencedInETC.html.

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D-Triangle

Cite this as:

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

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