First Droz-Farny Circle

Author

Eric W. Weisstein
January 16, 2005

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

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

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

Circles centered at vertices cutting midpoint triangle (generalization of Droz-Farny circle)

tri={{0,0},{1,0},{.45,.9}};
r=.46;
o=Orthocenter[tri][[1]];
m=Take[MidpointTriangle[tri][[1]],3];
p=DrozFarnyPoints[tri,r];
pts={Orthocenter,"H",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])",
	p[[1,1]],"(Subscript[P, 1])",p[[1,2]],"(Subscript[Q, 1])",p[[2,1]],"(Subscript[P, 2])",p[[2,2]],"(Subscript[Q, 2])",
		p[[3,1]],"(Subscript[P, 3])",p[[3,2]],"(Subscript[Q, 3])"};

Show[Graphics[{
  {Thickness[.01],Triangle[tri]},
  PointLabels[tri,pts,.03,{.02,.04}],
  DrozFarnyVertexCircle[tri,r],
  MidpointTriangle[tri],DrozFarnyLines[tri,r],
			{Dashing[{.02,.02}],Table[Circle[tri[[i]],r],{i,3}]}
}],AspectRatio->Automatic,PlotRange->{{-.2,1.2},{-.2,1.}}]

vertex-circumcenter Droz-Farny circle

tri={{0,0},{1,0},{.4,.9}};
r=Circumradius[tri];
o=Orthocenter[tri][[1]];
m=Take[MidpointTriangle[tri][[1]],3];
p=DrozFarnyPoints[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])",
	p[[1,1]],"(Subscript[P, 1])",p[[1,2]],"(Subscript[Q, 1])",p[[2,1]],"(Subscript[P, 2])",p[[2,2]],"(Subscript[Q, 2])",
		p[[3,1]],"(Subscript[P, 3])",p[[3,2]],"(Subscript[Q, 3])"};

Show[Graphics[{
  {Thickness[.007],Triangle[tri]},
  PointLabels[tri,pts,.03,{.02,.04}],
  {Thickness[.01],DrozFarnyOrthocircle[tri]},
			Table[Line[{p[[i,1]],p[[i,2]]}],{i,3}],
			{Dashing[{.02,.02}],Circumcircle[tri],Table[Circle[tri[[i]],r],{i,3}]}
}],AspectRatio->Automatic,PlotRange->{{-.2,1.2},{-.4,1.}}]

Altitude feet-circumcenter (First) Droz-Farny circle

Graphics

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

Show[Graphics[{
	{Thickness[.007],Triangle[tri]},
  {Dashing[{.02,.02}],Circumcircle[tri]},
  {GrayLevel[.5],Altitudes[tri]},
	PointLabels[tri,pts,.03,{.03,.05}],
	{Thickness[.01],DrozFarnyOrthocircle[tri]},
  {Dashing[{.02,.02}],
				Table[Circle[a[[i,1]],Sqrt[(a[[i,1]]-c).(a[[i,1]]-c)]],{i,1,3}]}
}],AspectRatio->Automatic]

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

Show[Graphics[{
  {Thickness[.007],Triangle[tri]},
  {GrayLevel[.5],Altitudes[tri]},
  PointLabels[tri,pts,.03,{.02,.04}],
  {Thickness[.01],DrozFarnyOrthocircle[tri]},
  {Dashing[{.02,.02}],
				Table[Circle[a[[i,1]],Sqrt[(a[[i,1]]-c).(a[[i,1]]-c)]],{i,1,3}]},
			Table[Line[{p[[i,1]],p[[i,2]]}],{i,3}],
			{Dashing[{.02,.02}],Circumcircle[tri],Table[Circle[tri[[i]],r],{i,3}]}
}],AspectRatio->Automatic,PlotRange->{{-.2,1.2},{-.4,1.}}]

Derivation

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

Circle Function

Factor[CircleFunction[eqn]]

InputForm[%]

CircleFunction[DrozFarnyCircle[1]]

IncidentCenters[KimberlingCenters,CyclicTrilinears[Symmetrize[CircleFunction[DrozFarnyCircle[1]]]]]//Timing

Center

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

Symmetrize[LawOfCosines[%]]

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

CenterName[X[4]]

Radius

r=Together//@DistanceTrilinears[CircleRadius[DrozFarnyCircle[1]]]

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

(1/((Sqrt[2]) ))((Sqrt[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[1]]//Timing