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# D-Triangle

Let the circles and used in the construction of the Brocard points which are tangent to at and , respectively, meet again at . The points then define the -triangle, also known as the fourth Brocard triangle (Gibert).

It has trilinear vertex matrix

 (1)

The vertices of the -triangle are the isogonal conjugates of the second Brocard triangle, and is inversely similar to the medial triangle (Johnson 1929, p. 285). In addition, 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

 (2) (3) (4)

(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 .

 center of the D-triangle center of reference triangle circumcenter midpoint of and symmedian point symmedian point first isodynamic point first Fermat point second isodynamic point second Fermat point far-out point Parry point Parry point triangle centroid Schoute center center of Kiepert hyperbola isogonal conjugate of direction of vector , where isogonal conjugate of crossdifference of line and Collings transform of orthocenter radical center of (circumcircle, Brocard circle, Parry circle) tripolar triangle centroid of isogonal conjugate of point Biham isogonal conjugate of isogonal conjugate of

Apollonius Circle, Brocard Points, Second Brocard Triangle, Third Brocard Triangle

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## References

Gibert, B. "Brocard Triangles." http://perso.wanadoo.fr/bernard.gibert/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.

D-Triangle

## Cite this as:

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