I would like to share an interesting way to generate radical axis for two circles given in the title by GeometricScene
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radical line
MATHWORLD TERM
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The radical line, also called the radical axis, is the locus of points of equal circle power with respect to two nonconcentric circles. By the chordal theorem, it is perpendicular to the line of centers.
Let the circles have radii
r
1
and
r
2
and their centers be separated by a distance
d
. If the circles intersect in two points, then the radical line is the line passing through the points of intersection. If not, then draw any two circles which cut each original circle twice. Draw lines through each pair of points of intersection of each circle. The line connecting their two points of intersection is then the radical line.
Given two circles with trilinear equations
(lα+mβ+nγ)(aα+bβ+cγ)​​+k(aβγ+bγα+cαγ)0​​(
′
l
α+
′
m
β+
′
n
γ)(aα+bβ+cγ)​​+
′
k
(aβγ+bγα+cαβ)0,
their radical line has equation
(
′
k
l-k
′
l
)α+(
′
k
m-k
′
m
)β+(
′
k
n-k
′
n
)γ0
(Kimberling 1998, p. 224).
Usually the procedural (compass and ruler version) to find the radical axis for two fixed detached circle O1 and O2 are the following: introduce a new circle O3 and find two sets of intersections, {P1,P2} and {Q1,Q2}.
The two secant lines meet at
P
. Do this again with smaller circle centered at O3 and find a new
P'
. By joining
P
and

P
′ we have found the radical axis.
We can take advantage of RandomIntance to use tangent length as result of circle power of point:
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gs=GeometricScene[{{"P1","P2","P","O1"->{0,0},"O2"->{7,0},"O3"},{r}},{cir1=Circle["O1",2],cir2=Circle["O2",3],GeometricAssertion[{cir1,Circle["O3",r]},{"Tangent","P1"}],GeometricAssertion[{Line[{"P","P1"}],Circle["O3",r]},{"Tangent","P1"}],GeometricAssertion[{cir2,Circle["O3",r]},{"Tangent","P2"}],GeometricAssertion[{Line[{"P","P2"}],Circle["O3",r]},{"Tangent","P2"}]}];​​instance=RandomInstance[gs,2,RandomSeeding->1001]
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radicalLinePts="P"/.Through[instance["Points"]]
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{{3.14286,-0.358263},{3.14286,-3.76018}}
We can visualize the tangent rays shooting from any point on the radical axis:
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