Brocard circle

In geometry, the Brocard circle (or seven-point circle) for a triangle is a circle defined from a given triangle. It passes through the circumcenter and symmedian 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 ${\displaystyle a}$, ${\displaystyle b}$, and ${\displaystyle c}$ of the given triangle, and the areal coordinates ${\displaystyle (x,y,z)}$ for points inside the triangle (where the ${\displaystyle x}$-coordinate of a point is the area of the triangle made by that point with the side of length ${\displaystyle a}$, etc), the Brocard circle consists of the points satisfying the equation[1]

${\displaystyle a^{2}yz+b^{2}zx+c^{2}xy={\frac {a^{2}b^{2}c^{2}(x+y+z)}{a^{2}+b^{2}+c^{2}}}\left({\frac {x}{a^{2}}}+{\frac {y}{b^{2}}}+{\frac {z}{c^{2}}}\right).}$

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

1. 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
2. Cajori, Florian (1917), A history of elementary mathematics: with hints on methods of teaching, The Macmillan company, p. 261.
3. Honsberger, Ross (1995), Episodes in Nineteenth and Twentieth Century Euclidean Geometry, New Mathematical Library, 37, Cambridge University Press, p. 110, ISBN 9780883856390.
4. Smart, James R. (1997), Modern Geometries (5th ed.), Brooks/Cole, p. 184, ISBN 0-534-35188-3
5. Guggenbuhl, Laura (1953), "Henri Brocard and the geometry of the triangle", The Mathematical Gazette, 37 (322): 241–243, JSTOR 3610034.
6. 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.

See also

• Nine-point circle