Adams' Circle

Author

Eric W. Weisstein
February 12, 2004

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

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

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

Plot

tri=Triangle[{{0.,0.},{1.1,2.},{3.,0.}}];
v=Vertices@tri;
adam=AdamsCirclePoints[tri];
ct=ContactTriangle[tri];
pts={v[[1]],"A",v[[2]],"B",v[[3]],"C",
GergonnePoint,"Ge",Incenter,"I",
ct[[1,1]],"(Subscript[T, A])",ct[[1,2]],"(Subscript[T, B])",ct[[1,3]],"(Subscript[T, C])",
adam[[1]],"U",adam[[2]],"S",adam[[3]],"Q",
adam[[4]],"T",adam[[5]],"R",adam[[6]],"P"};

Show[Graphics[{
tri,
ct,
AdamsCircle[tri],
PointLabels[tri,pts,.025,{.0,-.1}],
Table[Line[{GergonnePoint[tri][[1]],adam[[i,1]]}],{i,6}],
{Dashing[{.02,.02}],Incircle[tri],Table[Line[{ct[[1,i]],v[[i]]}],{i,3}]}
}],AspectRatio->Automatic]

t=Triangle[{{0,0},{1,0},{.3,.5}}];
circ=Circle[Incenter[t][[1]],(1/(Sqrt[(((a+b+c) ((((a^2)+(((b-c))^2)-2 a (b+c)))^2))/((a-b-c) (a+b-c) (a-b+c) (b (((b-c))^2) c+(a^3) (b+c)+a (((b-c))^2) (b+c)-(a^2) (2 (b^2)+b c+2 (c^2)))))]))]/.Thread[{a,b,c}->SideLengths[Evaluate[t]]];
circ2=AdamsCircle[t];
pts=Intersections[circ2,#]&/@Sides[t];
Show[Graphics[{
t,
circ2,
ContactTriangle[t],
PointSize[.02],pts,GergonnePoint[t],Incenter[t]
}],AspectRatio->Automatic]

Radius

pl=ParallelLine[Trilinears[GergonnePoint],TrilinearLine[Drop[TrilinearVertexMatrix[ContactTriangle],-1]]]

r=FullSimplify[Intersections[pl,{{1,0,0},{0,0,1}}]]

rr=FullSimplify[Together//@Distance[Trilinears[Incenter],r],TriangleQ[]]

q=FullSimplify[Intersections[pl,{{0,1,0},{0,0,1}}]]

rq=FullSimplify[Together//@Distance[Trilinears[Incenter],q],TriangleQ[]]

rr==rq

Refine[rq,TriangleQ[]]

2Inradius[]

Simplify[Sqrt[- (b (((b-c))^2) c+(a^3) (b+c)+a (((b-c))^2) (b+c)-(a^2) (2 (b^2)+b c+2 (c^2)))]/(-(a^2)-(((b-c))^2)+2 a (b+c)) (Sqrt[((-(a-b-c) (a+b-c) (a-b+c))/(a+b+c))]) r/Inradius[],TriangleQ[]]

ExpandAll[%]

ra=-(2 √(-(a^3) b+2 (a^2) (b^2)-a (b^3)-(a^3) c+(a^2) b c+a (b^2) c-(b^3) c+2 (a^2) (c^2)+a b (c^2)+2 (b^2) (c^2)-a (c^3)-b (c^3)) r)/((a^2)-2 a b+(b^2)-2 a c-2 b c+(c^2));

t=N[Triangle[{{0,0},{1,0},{7/20,6/10}}],100];

ra/.r->Inradius[]/.Thread[{a,b,c}->SideLengths[Evaluate[t]]]

((r (Sqrt[(p^2)-a b c s-p (s^2)]))/(p-(s^2)))/.r->Inradius[]/.s->Semiperimeter[]/.p->(a b+b c+c a)/.Thread[{a,b,c}->SideLengths[Evaluate[t]]]

Factor[eqn=-(a^3) b+2 (a^2) (b^2)-a (b^3)-(a^3) c+(a^2) b c+a (b^2) c-(b^3) c+2 (a^2) (c^2)+a b (c^2)+2 (b^2) (c^2)-a (c^3)-b (c^3)]

Factor[eqn2=(a^2)-2 a b+(b^2)-2 a c-2 b c+(c^2)]

cos=CyclicTrilinears[Numerator[Together[#-DistanceTrilinears[#]&[Cos[A]Cos[B]Cos[C]]]]]

Factor[GroebnerBasis[{eqn2,Sequence@@cos},{a,b,c},MonomialOrder->EliminationOrder]]

Factor[GroebnerBasis[{eqn,Sequence@@cos},{a,b,c},MonomialOrder->EliminationOrder]]

sin=CyclicTrilinears[Numerator[Together[#-DistanceTrilinears[#]&[(Sin[A]Sin[B]Sin[C])^2]]]]

Factor[GroebnerBasis[{eqn,Sequence@@sin},{a,b,c},MonomialOrder->EliminationOrder]]

<<Algebra`SymmetricPolynomials`

SymmetricReduction[#,{a,b,c}]&/@{eqn,eqn2}

%/.a+b+c:>2s/.(a b+a c+b c)->p

Simplify[2Sqrt[4 (p^2)-4 a b c s-4 p (s^2)]r/(4s^2-4p)]

Rrs=Numerator[Together[{Circumradius[]^2-R^2,Inradius[]^2-r^2,Semiperimeter[]-s}]]

Factor[GroebnerBasis[{eqn2,Sequence@@Rrs},{a,b,c},MonomialOrder->EliminationOrder]]

Factor[Together//@Expand[((r^2) (r-4 R) (r+4 R) (s^2)/.R->Circumradius[]/.r->Inradius[]/.s->Semiperimeter[])]]

{eqn,eqn2}

Factor[GroebnerBasis[{eqn,Sequence@@Rrs},{a,b,c},MonomialOrder->EliminationOrder]]

eqn

Factor[FullSimplify[Together//@(-(r^2) (s^2) ((r^2)-8 r R+16 (R^2)+(s^2)) ((r^2)+8 r R+16 (R^2)+(s^2))/.R->Circumradius[]/.r->Inradius[]/.s->Semiperimeter[]),TriangleQ[]]]

Factor[FullSimplify[Together//@((r^2) (s^2) (2 (b^2)+2 b c+2 (c^2)+(r^2)-16 (R^2)-4 b s-4 c s+(s^2))/.R->Circumradius[]/.r->Inradius[]/.s->Semiperimeter[]),TriangleQ[]]]

eqn

Together//@Expand[-(r^2) (s^2) (2 (b^2)+2 b c+2 (c^2)+(r^2)-16 (R^2)-4 b s-4 c s+(s^2))16/(a+b+c)^2/.R->Circumradius[]/.r->Inradius[]/.s->Semiperimeter[]]

FullSimplify[-2r Sqrt[-(r^2) (s^2) (2 (b^2)+2 b c+2 (c^2)+(r^2)-16 (R^2)-4 b s-4 c s+(s^2))16/(a+b+c)^2],TriangleQ[]&&r>0&&s>0]/.(a+b+c)->2s

(Sqrt[Expand[-2 ((b^2)+b c+(c^2))-(r^2)+16 (R^2)+4 (b+c) s-(s^2)]])

((-4r^2(Sqrt[-2 (b^2)-2 b c-2 (c^2)-(r^2)+16 (R^2)+4 b s+4 c s-(s^2)]))/((a^2)-2 a b+(b^2)-2 a c-2 b c+(c^2)))

Factor[Together[Expand[Inradius[]^2]]/((a^2)-2 a b+(b^2)-2 a c-2 b c+(c^2))]

((-4r^2(Sqrt[-2 (b^2)-2 b c-2 (c^2)-(r^2)+16 (R^2)+4 b s+4 c s-(s^2)]))/((a^2)-2 a b+(b^2)-2 a c-2 b c+(c^2)))

eqn

eqn

SymmetricReduction[#,{a,b,c},Array[s,3]]&/@{-(a^3) b+2 (a^2) (b^2)-a (b^3)-(a^3) c+(a^2) b c+a (b^2) c-(b^3) c+2 (a^2) (c^2)+a b (c^2)+2 (b^2) (c^2)-a (c^3)-b (c^3),(a^2)-2 a b+(b^2)-2 a c-2 b c+(c^2)}

((-2 r Sqrt[-((s[1])^2) s[2]+4 ((s[2])^2)-2 s[1] s[3]])/(((s[1])^2)-4 s[2]))

((2 r (Sqrt[(4 s[2]-((s[1])^2) )s[2]-2 s[1] s[3]]))/(4 s[2]-((s[1])^2)))

<<Algebra`SymmetricPolynomials`

Circle Function

f=FullSimplify[Together//@DistanceTrilinears[CircleFunction[CircleEquation[Trilinears[Incenter],1/(√(((a+b+c) ((((a^2)+(((b-c))^2)-2 a (b+c)))^2))/((a-b-c) (a+b-c) (a-b+c) (b (((b-c))^2) c+(a^3) (b+c)+a (((b-c))^2) (b+c)-(a^2) (2 (b^2)+b c+2 (c^2))))))]]],TriangleQ[]]

Factor[%]

CircleFunction[AdamsCircle]//InputForm

IncidentCenters[KimberlingCenters,CyclicTrilinears[CircleFunction[AdamsCircle]]]//Timing

Center

Together//@DistanceTrilinears[CircleCenter[AdamsCircle]]

Equal@@%

Plot

t=Triangle[Table[Random[Real,{0,1},60],{3},{2}]];

circ=Circle[Incenter[t][[1]],(1/(Sqrt[(((a+b+c) ((((a^2)+(((b-c))^2)-2 a (b+c)))^2))/((a-b-c) (a+b-c) (a-b+c) (b (((b-c))^2) c+(a^3) (b+c)+a (((b-c))^2) (b+c)-(a^2) (2 (b^2)+b c+2 (c^2)))))]))]/.Thread[{a,b,c}->SideLengths[Evaluate[t]]];
pts=Intersections[circ,#]&/@Sides[t];
Distance[Incenter[t],#]&/@Flatten[pts]

Incicent Centers

IncidentCenters[KimberlingCenters,AdamsCircle]//Timing