H4_pic0.2.(alternative)

import geometry;

size(9cm, 0);
defaultpen(fontsize(10pt));
dotfactor = 4;

point A,B,C,O1,O2,O3;
A=origin; B= 5*dir(60); C=5*dir(0);
triangle T=triangle(A,B,C);
point A1 = bisectorpoint(T.BC);
point B1 = bisectorpoint(T.AC);

circle incirc1=incircle(A,B,B1); O1=incirc1.C;
circle incirc2=incircle(A,C,A1); O2=incirc2.C;
circle incirc3=incircle(A,B,A1); O3=incirc3.C;

point D = projection(A,C)*O1;
point Ep = projection(A,C)*O2;


draw(O1--D^^O2--Ep);
draw(A--O1^^C--O2);
draw(O1--O2--O3--cycle);
draw(incirc1^^incirc2^^incirc3);
draw(T); label(T);