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Brocard circle

Circle constructed from a triangle

Ingeometry, the Brocard circle (orseven-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[edit]

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]

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

Related points[edit]

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[edit]

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

History[edit]

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[edit]

External links[edit]

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

See also[edit]

• Nine-point circle