Residual triangles after incircle placement

Question

After calculating the incircle for a triangle, I would like to get the vertex coordinates of the three left-over triangular regions. The additional constraint is that the angle bisectors are at right angles with the yellow lines shown.

SeedRandom[2];
tri = RandomPolygon[{"Convex", 3}]
c = TriangleCenter[tri[[1]], "Incenter"];
r = TriangleMeasurement[tri[[1]], "Inradius"];
d1 = Disk[c, r];

Graphics[{tri, Red, d1
  }
 , Frame -> True
 ]

Thanks for your help.

Answer 1

SeedRandom[2];
tri = RandomPolygon[{"Convex", 3}];
c = TriangleCenter[tri[[1]], "Incenter"];
r = TriangleMeasurement[tri[[1]], "Inradius"];
d1 = Disk[c, r];
{a1, b1, c1} = tri[[1]];
a2 = RegionIntersection[Line[{a1, c}], Circle[c, r]][[1, 1]];
b2 = RegionIntersection[Line[{b1, c}], Circle[c, r]][[1, 1]];
c2 = RegionIntersection[Line[{c1, c}], Circle[c, r]][[1, 1]];
Graphics[{tri, Red, d1, Yellow, 
  Point@c, {Yellow, 
   RegionIntersection[InfiniteLine[a2, Cross[a1 - c]], 
     RegionBoundary@tri] /. Point -> Line, 
   RegionIntersection[InfiniteLine[b2, Cross[b1 - c]], 
     RegionBoundary@tri] /. Point -> Line, 
   RegionIntersection[InfiniteLine[c2, Cross[c1 - c]], 
     RegionBoundary@tri] /. Point -> Line}}, Frame -> True]