Brocard Axis

Author

Eric W. Weisstein
February 6, 2005

This notebook downloaded from http://mathworld.wolfram.com/notebooks/PlaneGeometry/BrocardAxis.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/BrocardAxis.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Figure

t=Triangle[{{0.,0.},{.6,1.4},{3.,-.1}}];
IsodynamicPoint1[t_]:=IsodynamicPoints[t][[1]];
IsodynamicPoint2[t_]:=IsodynamicPoints[t][[2]];
pts={
LemoinePoint,K,
Circumcenter,O,
IsodynamicPoint1,S,
IsodynamicPoint2,S',
BrocardMidpoint,(Subscript[M, B]),
};

Show[Graphics[
{
t,
PointLabels[t,pts],
LongestLine[Coordinates/@PointPoints[t,pts]]
}],
AspectRatio->Automatic,PlotRange->All]

Equation

LineEquation[BrocardAxis]

Incident Points

CollinearQ[{Circumcenter,SymmedianPoint,IsodynamicPoint[1],IsodynamicPoint[2],BrocardMidpoint}]

(ctrs=IncidentCenters[KimberlingCenters,BrocardAxis])//Timing

First/@ctrs

Length[ctrs]

{#,CenterName[#]}&/@ctrs//TableForm

Check

ctrs={X[3],X[6],X[15],X[16],X[32],X[39],X[50],X[52],X[58],X[61],X[62],X[182],X[187],X[216],X[284],X[371],X[372],X[386],X[389],X[500],X[511],X[800],X[970],X[991],X[1030],X[1151],X[1152],X[1160],X[1161],X[1326],X[1333],X[1342],X[1343],X[1350],X[1351],X[1384],X[1504],X[1505],X[1570],X[1578],X[1579],X[1670],X[1671],X[1687],X[1688],X[1689],X[1690],X[1691],X[1692],X[1970],X[1983],X[2021],X[2024],X[2025],X[2030],X[2031],X[2032],X[2035],X[2036],X[2076],X[2080],X[2088],X[2092],X[2458],X[2459],X[2460],X[2461],X[2462],X[2673],X[2674]};

Simplify[DistanceTrilinears[TrigExpand[LineEquation[BrocardAxis]/.Thread[{α,β,γ}->Trilinears[#]]]],TriangleQ[]]&/@ctrs//Timing