In[]:=
a={1,Sqrt[3]};b={0,0};c={2,0};
In[]:=
BE/BA = λ1 , CD/CA = λ2
In[]:=
inradiusAEC=FunctionInterpolation[​​TriangleMeasurement[{λ1*(a-b)+b,a,c},"Inradius"],{λ1,0.01,0.99}];​​inradiusABD=FunctionInterpolation[​​TriangleMeasurement[{λ2*(a-c)+c,a,b},"Inradius"],{λ2,0.01,0.99}];​​findLambda2[a_?ListQ,λ1_?NumericQ]:=Block[{λ2},λ2/.FindRoot[inradiusAEC[λ1]inradiusABD[λ2],{λ2,0.5},PrecisionGoal4,AccuracyGoal4]]
In[]:=
segmentIntersection[p1_List,p2_List,p3_List,p4_List]:=Block[{u,v},Module[{sol},sol=Solve[u*(p2-p1)+p1v*(p4-p3)+p3,{u,v}];​​{u,v,u*(p2-p1)+p1}/.sol〚1〛]]
In[]:=
In[]:=
findP[a_?ListQ,λ1_?NumericQ]:=With[{λ2=findLambda2[a,λ1]},segmentIntersection[b,λ2*(a-c)+c,c,λ1*(a-b)+b]][[-1]]
In[]:=
sol=FindRoot[inradiusAEC[λ1]​​TriangleMeasurement[{findP[a,λ1],b,c},"Inradius"],{λ1,0.5},PrecisionGoal4,AccuracyGoal4]
Out[]=
{λ10.579796}
In[]:=
TriangleMeasurement[{findP[a,λ1],b,c}/.sol,"Inradius"]
Out[]=
0.317837