Brocard circle

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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).

The two Brocard points lie on this circle, as do the vertices of the Brocard triangle.[1] It is concentric with the first Lemoine circle.[2]

If the triangle is equilateral, the circumcenter and symmedian coincide and therefore the Brocard circle reduces to a single point.[3]

The Brocard circle is named for Henri Brocard,[4] who presented a paper on it to the French Association for the Advancement of Science in Algiers in 1881.[5]


  1. Cajori, Florian (1917), A history of elementary mathematics: with hints on methods of teaching, The Macmillan company, p. 261<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>.
  2. Honsberger, Ross (1995), Episodes in Nineteenth and Twentieth Century Euclidean Geometry, New Mathematical Library, 37, Cambridge University Press, p. 110, ISBN 9780883856390<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>.
  3. Smart, James R. (1997), Modern Geometries (5th ed.), Brooks/Cole, p. 184, ISBN 0-534-35188-3<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
  4. Guggenbuhl, Laura (1953), "Henri Brocard and the geometry of the triangle", The Mathematical Gazette, 37 (322): 241–243, JSTOR 3610034<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>.
  5. O'Connor, John J.; Robertson, Edmund F., "Henri Brocard", MacTutor History of Mathematics archive, University of St Andrews<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>.

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See also