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Circle Tangent to Circle, Given Center

tancir1.html

Exploration

Show that the radii of the two circles centered at "tancir1_1.gif" and tangent to the circle

"tancir1_2.gif"

are given by

"tancir1_3.gif"

where

"tancir1_4.gif".

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

Approach

Fix the center point using the equations "tancir1_5.gif" and "tancir1_6.gif". The circles are tangent if

"tancir1_7.gif"

where "tancir1_8.gif". 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

"tancir1_9.gif"

Assuming d>0 and "tancir1_10.gif": (1) "tancir1_11.gif" is always negative, hence invalid; (2) "tancir1_12.gif" is positive if "tancir1_13.gif", i.e. "tancir1_14.gif" is outside circle "tancir1_15.gif"; (3) "tancir1_16.gif" is positive if "tancir1_17.gif", i.e. "tancir1_18.gif" is inside circle "tancir1_19.gif"; and (4) "tancir1_20.gif" is always positive and valid.


Copyright © 1999-2007 Donald L. Vossler, Descarta2D Publishing
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