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# Brocard Axis

The Brocard axis is the line passing through the symmedian point and circumcenter of a triangle, where the segment is the Brocard diameter (Kimberling 1998, p. 150). The Brocard axis is perpendicular to the Lemoine axis and is the isogonal conjugate of the Kiepert hyperbola.

It is central line and has equivalent trilinear equations

and

It passes through Kimberling centers for (circumcenter ), 6 (symmedian point ), 15 (first isodynamic point ), 16 (second isodynamic point ), 32 (third power point), 39 (Brocard midpoint), 50, 52, 58, 61, 62, 182, 187, 216, 284, 371 (Kenmotu point), 372, 386, 389 (Taylor center), 500, 511, 566, 567, 568, 569, 570, 571, 572, 573, 574, 575, 576, 577, 578, 579, 580, 581, 582, 583, 584, 800, 970, 991, 1030, 1151, 1152, 1160, 1161, 1182, 1192, 1207, 1326, 1333, 1340, 1341, 1342, 1343, 1350, 1351, 1379, 1380, 1384, 1504, 1505, 1570, 1578, 1579, 1609, 1620, 1662, 1663, 1664, 1665, 1666, 1667, 1668, 1669, 1670, 1671, 1683, 1684, 1685, 1686, 1687, 1688, 1689, 1690, 1691, 1692, 1693, 1694, 1805, 1806, 1970, 1983, 2011, 2012, 2019, 2020, 2021, 2022, 2024, 2025, 2026, 2027, 2028, 2029, 2030, 2031, 2032, 2033, 2034, 2035, 2036, 2037, 2038, 2055, 2076, 2080, 2088, 2092, 2193, 2220, 2245, 2271, 2278, 2305, 2420, 2456, 2458, 2459, 2460, 2461, 2462, 2558, 2559, 2560, 2561, 2562, 2563, 2673, 2674, 2965, 3001, 3002, 3003, and 3053.

Note that the Brocard axis is not equivalent to the Brocard line.

Brocard Circle, Brocard Diameter, Brocard Line, Brocard Triangles

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## References

Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.

Brocard Axis

## Cite this as:

Weisstein, Eric W. "Brocard Axis." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BrocardAxis.html