Second Droz-Farny Circle

Author

Eric W. Weisstein
January 16, 2005

This notebook downloaded from http://mathworld.wolfram.com/notebooks/PlaneGeometry/SecondDroz-FarnyCircle.nb.

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

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

Midpoint-Orthocenter (Second) Droz-Farny circle

Graphic

tri={{0,0},{1,0},{.4,.9}};
m=Midpoints[tri] /. Point->Identity;
o=Orthocenter[tri][[1]];
a=AltitudeFeet[tri];
p=DrozFarnyCircumpoints[tri];
pts={Orthocenter,"H",Circumcenter,"O",
		m[[1]],"(Subscript[M, 3])",m[[2]],"(Subscript[M, 1])",m[[3]],"(Subscript[M, 2])",
	tri[[1]],"(Subscript[A, 1])",tri[[2]],"(Subscript[A, 2])",tri[[3]],"(Subscript[A, 3])",
	a[[1]],"(Subscript[H, 1])",a[[2]],"(Subscript[H, 2])",a[[3]],"(Subscript[H, 3])",
			p[[1,1]],"(Subscript[P, 3])",p[[1,2]],"(Subscript[Q, 3])",p[[2,1]],"(Subscript[P, 1])",p[[2,2]],"(Subscript[Q, 1])",
		p[[3,1]],"(Subscript[P, 2])",p[[3,2]],"(Subscript[Q, 2])"};

Show[Graphics[{
			  {Thickness[.008],Triangle[tri]},
  {Dashing[{.02,.02}],Circumcircle[tri]},
  {Thickness[.01],DrozFarnyCircumcircle[tri]},
  {GrayLevel[.5],Altitudes[tri]},
			PointLabels[tri,pts,.03,{.03,.05}],
  {Dashing[{.02,.02}],
				Table[Circle[m[[i]],Sqrt[(m[[i]]-o).(m[[i]]-o)]],{i,1,3}]}
}],AspectRatio->Automatic]

Derivation

eqn=Symmetrize[Factor[Simplify[Together//@DistanceTrilinears[CircleEquation[Circumcenter,Sqrt[5Circumradius[]^2-(a^2+b^2+c^2)/2]]],TriangleQ[]]]]

Circle Function

LawOfCosines[Factor[CircleFunction[eqn]]]

DistanceTrilinears[%]

InputForm[%]

CircleFunction[DrozFarnyCircle[2]]

Symmetrize[CircleFunction[DrozFarnyCircle[2]]]

CircleFunction[DrozFarnyCircle[2]]:=((a^2+b^2-c^2)*(a^2-b^2+c^2)*(-a^2+b^2+c^2))/(2*b*(a-b-c)*(a+b-c)*c*(a-b+c)*(a+b+c))

IncidentCenters[KimberlingCenters,CyclicTrilinears[Symmetrize[CircleFunction[DrozFarnyCircle[2]]]]]//Timing

CenterName[X[6]]

Center

Symmetrize[Together//@DistanceTrilinears[CircleCenter[DrozFarnyCircle[2]][[1]]]]

Symmetrize[LawOfCosines[%]]

IncidentCenters[KimberlingCenters,CyclicTrilinears[Cos[A]]]//Timing

CenterName[X[3]]

Radius

r=Simplify[Factor[Together//@DistanceTrilinears[CircleRadius[DrozFarnyCircle[2]]]],TriangleQ[]]

Factor[r]

Factor[FullSimplify[Together//@Sqrt[5Circumradius[]^2-(a^2+b^2+c^2)/2],TriangleQ[]]]

(1/((Sqrt[2]) ))(√Factor[((((a^2)-(b^2)))^2) ((a^2)+(b^2))-((a^4)-4 (a^2) (b^2)+(b^4)) (c^2)-((a^2)+(b^2)) (c^4)+(c^6)])/((Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)]))

Incident Centers

IncidentCenters[KimberlingCenters,DrozFarnyCircle[2]]//Timing

Orthogonality

Check

Simplify[Together//@DistanceTrilinears[First[OrthogonalQ[PolarCircle,DrozFarnyCircle[2]]]],TriangleQ[]]//Timing

Graphic

t=Triangle[{{0,0},{1,0},{-.2,.7}}];
circs={PolarCircle,DrozFarnyCircle[2]};
centers={"(Subscript[X, 351])","(Subscript[X, 3])"};
colors={Red,Green};
Show[Graphics[{
Transpose[{colors,Through[circs[t]]}],
PointLabels[Join[
Transpose[{CircleCenter[#[t]]&/@circs,centers,colors}]
],LabelOffset->.1
]
}],AspectRatio->Automatic,PlotRange->All]