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Feuerbach Triangle


FeuerbachTriangle

The Feuerbach triangle is the triangle formed by the three points of tangency of the nine-point circle with the excircles (Kimberling 1998, p. 158). (The fact that the excircles touch the nine-point circle is known as Feuerbach's theorem.)

The Feuerbach triangle has trilinear vertex matrix

 [-sin^2[1/2(B-C)] cos^2[1/2(C-A)] cos^2[1/2(A-B)]; cos^2[1/2(B-C)] -sin^2[1/2(C-A)] cos^2[1/2(A-B)]; cos^2[1/2(B-C)] cos^2[1/2(C-A)] -sin^2[1/2(A-B)]].

The circumcenter of the Feuerbach triangle is the nine-point center of the reference triangle.

If I and N are the incenter and nine-point center of a triangle DeltaABC and F is its Feuerbach point, then DeltaABC and its Feuerbach triangle are perspective, and the perspector is the harmonic conjugate of F with respect to the segment IN. Equivalently, the perspector is the internal similitude center of the incircle and the nine-point circle.


See also

Feuerbach Point, Feuerbach's Theorem, Nine-Point Circle

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References

Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.

Referenced on Wolfram|Alpha

Feuerbach Triangle

Cite this as:

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

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