The Brocard axis is the line 
passing through the symmedian
point 
and circumcenter 
of a triangle, where the line
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.
The Brocard axis contains the centers of four distinct circles associated with nine of the 36 cataloged six-piece
triangle dissections of a generic non-right
triangle into similar
triangles. In these nine layouts, all four boundary cut points
are concyclic. Three layouts with junctions at vertices
of the third Brocard triangle give three
distinct circles. The remaining six use the first
Lemoine circle, whose center is the midpoint
of 
(E. Pegg, Jr., pers. comm., Oct. 3,
2026).
Note that the Brocard axis is not equivalent to the Brocard line.
