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Brocard circle

Circle constructed from a triangle

Brocard circle

Circle constructed from a triangle

In geometry, the Brocard circle (or seven-point circle) is a circle derived from a given triangle. It passes through the circumcenter and symmedian point of the triangle, and is centered at the midpoint of the line segment joining them (so that this segment is a diameter).

Equation

In terms of the side lengths a, b, and c of the given triangle, and the areal coordinates (x,y,z) for points inside the triangle (where the x-coordinate of a point is the area of the triangle made by that point with the side of length a, etc), the Brocard circle consists of the points satisfying the equation{{citation | access-date = 2019-01-05 | archive-url = https://web.archive.org/web/20180422182635/http://forumgeom.fau.edu/FG2005volume5/FG200513.pdf | archive-date = 2018-04-22 | url-status = dead :b^2 c^2 x^2 + a^2 c^2 y^2 + a^2 b^2 z^2 - a^4 y z - b^4 x z - c^4 x y=0.

Special cases

If the triangle is equilateral, the circumcenter and symmedian coincide and therefore the Brocard circle reduces to a single point.

History

The Brocard circle is named for Henri Brocard,{{citation

References

References

  1. Smart, James R.. (1997). "Modern Geometries". Brooks/Cole.
  2. "Henri Brocard".
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