Let the circles c_2 and c_3^' used in the construction of the Brocard points which are tangent to A_2A_3 at A_2 and A_3, respectively, meet again at D_A. The points D_AD_BD_C then define the D-triangle, also known as the fourth Brocard triangle (Gibert).

It has trilinear vertex matrix

 [asecA 2c 2b; 2c bsecB 2a; 2b 2a csecC].

The vertices of the D-triangle are the isogonal conjugates of the second Brocard triangle, and DeltaD_AD_BD_C 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 D-triangle is the orthocentroidal circle, which has diameter GH, where G is the triangle centroid and H 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 X_n.

See also

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

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Gibert, B. "Brocard Triangles.", 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.

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Cite this as:

Weisstein, Eric W. "D-Triangle." From MathWorld--A Wolfram Web Resource.

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