Circle Tangent to Two Lines, Radius

tancir3.html

Exploration

Show that the centers (h,k) of the four circles tangent to the perpendicular lines

and

with radius r=1 are given by

,

,

,

.

Assume that the two lines are normalized, .

Approach

A circle tangent to a line A x+B y+C=0 implies that

giving two equations.  The fixed radius r=1 is a third equation. Solve three equations in three 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 three equations.

Clear[r,h,k,A1,B1];
ans1=Solve[{r^2==( A1*h+B1*k)^2,
            r^2==(-B1*h+A1*k)^2,
            r==1},
           {h,k,r}]

Simplify.

ans2=ans1 //. A1^2+B1^2->1