Incenter on Knot Circle

knotin.html

Exploration

Show that the incenter of a triangle (the center point of the circle inscribed in the triangle) is on one of the knot circles for the biarc configuration defined by the triangle.

Approach

Construct a triangle in a simplified position. Construct the incenter. Construct the knot circles. Show that the incenter is on one of the knot circles.

Initialize

To initialize Descarta2D, select the input cell bracket and press SHIFT-Enter.

This initialization assumes that the Descarta2D software has been copied into one of the standard directories for AddOns which are on the Mathematica search path, $Path.

<<Descarta2D`

Solution

Define a function to compute the knot circles.

KnotCircles2D[
   t1:Triangle2D[p1:{x1_,y1_},
                 pA:{xA_,yA_},
                 p2:{x2_,y2_}]] :=
Module[{pt1,pt2},
   pt1=Point2D[t1,Inscribed2D];
   pt2=Point2D[Point2D[pA],
               Point2D[p1],
               -Distance2D[p2,pA]];
   Map[Circle2D[Point2D[p1],
                Point2D[p2],#]&,
       {pt1,pt2}] ];       

Create the triangle.

Clear[a,b];
t1=Triangle2D[{0,0},{a,b},{1,0}];

Construct the incenter of the triangle

pt1=Point2D[t1,Inscribed2D] //Simplify

Construct knot circles for the triangle.

kc1=KnotCircles2D[t1][[1]] //FullSimplify

Show that the incenter is on the circle.

eq1=Polynomial2D[
       Quadratic2D[kc1],
       Coordinates2D[pt1]] //Simplify