Circle Tangent to Circle, Given Center

tancir1.html

Exploration

Show that the radii of the two circles centered at and tangent to the circle

are given by

where

.

This formula is a special case of TangentCircles2D[{pt | ln | cir}, point].

Approach

Fix the center point using the equations and . The circles are tangent if

where . Solve the three equations for r.

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 three equations.

Clear[h,h1,k,k1,d,r,r2];
ans1=Solve[{h==h1 &&
            k==k1,
            (d^2-(r2-r)^2)*(d^2-(r2+r)^2)==0},
           {h,k,r}] //Simplify

Assuming d>0 and : (1) is always negative, hence invalid; (2) is positive if , i.e. is outside circle ; (3) is positive if , i.e. is inside circle ; and (4) is always positive and valid.