Tarry Point

Author

Eric W. Weisstein
January 30, 2003

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

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

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

Graphic

tri={{0,0},{1,-0.1},{0.52,0.5}};
bt=BrocardTriangle[1,tri][[1]];
pts={bt[[1]],"(Subscript[B, 3])",bt[[2]],"(Subscript[B, 1])",bt[[3]],"(Subscript[B, 2])",
		tri[[1]],"(Subscript[A, 1])",tri[[2]],"(Subscript[A, 2])",tri[[3]],"(Subscript[A, 3])",
	BrocardPoint[1,tri],"Ω",BrocardPoint[2,tri],"Ω'",
	TarryPoint,"T",SteinerPoint,"S",Circumcenter,"O"};

Show[Graphics[{
		Triangle[tri],
		BrocardTriangle[1,tri],
		{GrayLevel[.5],BrocardLines[1,tri][[1]]},
		{GrayLevel[.5],Dashing[{.02,.02}],BrocardLines[1,tri][[2]]},
		Circumcircle[tri],
		Table[Line[{tri[[i]],TarryPoint[tri][[1]]}],{i,3}],
		PointLabels[tri,pts,.03,{.03,.03}],
			{Dashing[{.01,.04}],Line[{SteinerPoint[tri],TarryPoint[tri]} /. Point->Identity]}
}],AspectRatio->Automatic]

tri={{0.,0.},{1.,1.3},{3.,0.}};
bt=BrocardTriangle[1,tri][[1]];
pts={SteinerPoint,"S",TarryPoint,"T",Circumcenter,"O",
	tri[[1]],"A",tri[[2]],"B",tri[[3]],"C",
	bt[[1]],"",bt[[2]],"",bt[[3]],""};
t=TarryPoint[tri][[1]];
l=Table[Line[{tri[[i]],t}],{i,3}];

Show[Graphics[{
			{Dashing[{.02,.02}],l},
			{Dashing[{.01,.04}],Line[{SteinerPoint[tri],TarryPoint[tri]} /. Point->Identity]},
			Dashing[{}],
			Triangle[tri],Circumcircle[tri],
			BrocardTriangle[1,tri],
			PointLabels[tri,pts,.03]
}],AspectRatio->Automatic]

tri={{0.,0.},{1.,1.3},{3.,0.}};
bt=Take[BrocardTriangle[1,tri][[1]],3];
pts={SteinerPoint,"S",TarryPoint,"T",Circumcenter,"O",
	tri[[1]],"A",tri[[2]],"B",tri[[3]],"C",
	bt[[1]],"",bt[[2]],"",bt[[3]],"",TarryPoint[bt],"(Subscript[T, B])"};
t=TarryPoint[tri][[1]];

Show[Graphics[{
			Triangle[tri],Circumcircle[tri],
			BrocardTriangle[1,tri],
			PointLabels[tri,pts,.03]
}],AspectRatio->Automatic]

Center Function

FullSimplify[PowerExpand[Together//@(TrigExpand[FullSimplify[Sec[A+ArcCot[(a^2+b^2+c^2)/4/TriangleArea[{a,b,c}]]]]]/.Sin[A]:>Sqrt[1-Cos[A]^2]/.Cos[A]:>(b^2+c^2-a^2)/2/b/c)]]

Can drop constants and multiples symmetrical in (a,b ,c), so

((b c)/((b^4)+(c^4)-(a^2) ((b^2)+(c^2))))