Brocard circle

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^[1]

b^{2}c^{2}x^{2}+a^{2}c^{2}y^{2}+a^{2}b^{2}z^{2}-a^{4}yz-b^{4}xz-c^{4}xy=0.

Related points

The two Brocard points lie on this circle, as do the vertices of the Brocard triangle.^[2] These five points, together with the other two points on the circle (the circumcenter and symmedian), justify the name "seven-point circle".

The Brocard circle is concentric with the first Lemoine circle.^[3]

Special cases

If the triangle is equilateral, the circumcenter and symmedian coincide and therefore the Brocard circle reduces to a single point.^[4]

History

The Brocard circle is named for Henri Brocard,^[5] who presented a paper on it to the French Association for the Advancement of Science in Algiers in 1881.^[6]

References

^ Moses, Peter J. C. (2005), "Circles and triangle centers associated with the Lucas circles" (PDF), Forum Geometricorum, 5: 97–106, MR 2195737, archived from the original (PDF) on 2018-04-22, retrieved 2019-01-05
^ Cajori, Florian (1917), A history of elementary mathematics: with hints on methods of teaching, The Macmillan company, p. 261.
^ Honsberger, Ross (1995), Episodes in Nineteenth and Twentieth Century Euclidean Geometry, New Mathematical Library, vol. 37, Cambridge University Press, p. 110, ISBN 9780883856390.
^ Smart, James R. (1997), Modern Geometries (5th ed.), Brooks/Cole, p. 184, ISBN 0-534-35188-3
^ Guggenbuhl, Laura (1953), "Henri Brocard and the geometry of the triangle", The Mathematical Gazette, 37 (322): 241–243, doi:10.2307/3610034, JSTOR 3610034.
^ O'Connor, John J.; Robertson, Edmund F., "Henri Brocard", MacTutor History of Mathematics Archive, University of St Andrews

External links

Weisstein, Eric W. "Brocard Circle". MathWorld.