Circle Tangent to Circle, Center on Circle, Radius

tancir2.html

Exploration

Show that the centers (h,k) of the two circles passing through the point with center on the circle and radius r=1 are given by

where . This is a special case of TangentCircles2D[{obj},ln | cir,r], where the object is a point.

Approach

The radius is given, r=1, so the center point (h,k) needs to be found. The equation is formed noting that the given point is on the circle. The equation is formed noting that the center is on this circle. Solve two equations in two unknowns.

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

Solve the two equations.

Clear[h,k,x1,y1,d1];
ans1=Solve[{(x1-h)^2+(y1-k)^2==1, h^2+k^2==1},
           {h,k}] //. {
        x1^2+y1^2->d1^2} //FullSimplify

Simplify. Without loss of generality, assume all the point coordinates are positive.

Clear[E1];
ans2=ans1 //. {
   x1^2*y1+y1^3->y1*d1^2,
   x1^2+y1^2->d1^2,
   Sqrt[d1^2*E1_]->d1*Sqrt[E1],
   Sqrt[x1^2*E1_]->x1*Sqrt[E1]} //FullSimplify

ans3=ans2 //. {
   d1^3-d1*y1^2->d1*(d1^2-y1^2),
   d1^2-y1^2->x1^2}

ans4=Map[Apart,ans3,3]