(It is a long comment, not an answer) Last year, due to low accuracy of TikZ and its limitation for 3D figures, I wrote a plain Asymptote module for both 2D and 3D, named it ESgeometry.asy. Instead of using geometric constructions, it concentrated on using barycentric coordinates (signed area, signed length) and avaiable dot, cross products. Of course it included usual commands such as incenter, inradius, circumcenter, circumradius, .... I minimalized the number of commands by using only 2 main one: ESpoint and ESlength.
triple O=ESpoint("circumcenter",A,B,C,D);
real R=ESlength("circumradius",A,B,C,D);
The code has not been cleaned yet. I am going to do it soon.
// a part of ESgeometry.asy (for both 2D and 3D)
triple barycentric(triple A, triple B=(0,0,0), triple C=(0,0,0), triple D=(0,0,0), real a=1, real b=0, real c=0,real d=0){
return (a*A+b*B+c*C+d*D)/(a+b+c+d);
}
//////////////////////////////////////////////
// Compute signed area in 3D
real SignedArea(triple A, triple B, triple C){
return sgn(dot(cross(A,B),C))*abs(cross(B-A,C-A))/2;
}
// Compute (positive) area in 3D
real Area(triple A, triple B, triple C){
return abs(cross(B-A,C-A))/2;
}
triple ESpoint(string name="", triple A, triple B, triple C, triple D){
if (name=="circumcenter") {
triple O1=(abs(A-D))^2*cross(B-D,C-D);
triple O2=(abs(B-D))^2*cross(C-D,A-D);
triple O3=(abs(C-D))^2*cross(A-D,B-D);
real V=dot(A-D,cross(B-D,C-D));
return D+(O1+O2+O3)/(2*V);
}
else if (name=="incenter") {return barycentric(A,B,C,D,Area(B,C,D),Area(C,D,A),Area(D,A,B),Area(A,B,C));}
else if (name=="centroid") {return barycentric(A,B,C,D,1,1,1,1);}
else return (0,0,0);
}
triple ESpoint(string name="", triple A, triple B, triple C){
real a=abs(B-C), b=abs(C-A), c=abs(A-B);
if (name=="footbisector") {return barycentric(A,B,C,0,b,c);}
else if (name=="footaltitude") {return barycentric(A,B,C,0,1/(a^2+c^2-b^2),1/(a^2+b^2-c^2));}
else if (name=="incenter") {return barycentric(A,B,C,a,b,c);}
else if (name=="centroid") {return barycentric(A,B,C,1,1,1);}
else return (0,0,0);
}
///////////////////////////
unitsize(1cm);
import three;
import solids; // only for better sphere, Circle
triple B=(0,0,0), A=(5,-.5,2), C=(1,3.5,3), D=(1,1,6);
draw(A--B--C--cycle^^D--A^^D--B^^D--C);
label("$A$",A,plain.W);
label("$B$",B,plain.S);
label("$C$",C,plain.E);
label("$D$",D,plain.N);
triple I=ESpoint("incenter",A,B,C,D); dot(I,red);
triple Id=ESpoint("incenter",A,B,C); dot(Id,red);
triple I1=ESpoint("footbisector",A,B,C); dot(I1); draw(A--I1);
triple I2=ESpoint("footaltitude",Id,B,C); dot(I2); draw(Id--I2);
draw(Circle(Id,abs(Id-I2),normal=cross(A-B,A-C)));
real r=abs(I-(I+reflect(A,B,C)*I)/2);
revolution b=sphere(I,r);
draw(surface(b),yellow+opacity(.5));
// the foot of different bisectors
triple Fa=ESpoint("bisectorfoot",A,B,C); dot(Fa,red);
triple Fd=ESpoint("bisectorfoot",D,B,C); dot(Fd,blue);
triple circumcenter(triple A, triple B, triple C, triple D){
triple O1=(abs(A-D))^2*cross(B-D,C-D);
triple O2=(abs(B-D))^2*cross(C-D,A-D);
triple O3=(abs(C-D))^2*cross(A-D,B-D);
real V=dot(A-D,cross(B-D,C-D));
return D+(O1+O2+O3)/(2*V);
}
triple O=ESpoint("circumcenter",A,B,C,D);
revolution Oabcd=sphere(O,abs(O-D));
draw(surface(Oabcd),green+opacity(.3));
dot(A^^B^^C^^D,red);
triple M=barycentric(A,C,.5,.5);
draw(D--M);
triple Mg=barycentric(A,C,D,1,1,1);
dot(Mg,orange);
triple Atest=barycentric(B,1/2); draw(D--Atest);
//draw(shift(D)*surface(plane(A,C)),blue);
//draw(surface(A--B--C--cycle),blue+opacity(.5));
//draw(surface(B--C--D--cycle),red+opacity(.5));
//draw(surface(C--D--A--cycle),magenta+opacity(.5));
