$\textsf{\LARGE c}$

$\textsf{\LARGE A}$

Locus

When P moves on the circle c with center O and radius s, the intersection of the ray OP with the perpendicular bisector of AP
creates the locus of the equidistant points of c and A: a conic with semiaxis s and foci O and A.

If X is a point on the locus and A is on the circle, OP=XO+XA (ellipse) and if A is outside, OP=XO-XA (hyperbola branch).

s

Outside point - Circle

If A is inside the circle, we get an ellipse; if it is outside, a branch of a hyperbola.

Inside point - Circle

XA-Xc=0

I: (1, 0)
O = (0, 0)
r = Line(O, I)
H = Point(r)
B = Point(r)
s = 3 (slider from -5 to 5)
c = Circle(O, s)
P = Point(c)
A = Point(Segment[B, H])
I' = Mirror(I, O)
k = Circle(O, I)
op3 = false
h_1 = LineBisector(A, P)
i = Ray(O, P)
D_1 = Intersect(h_1, i)
lugar1 = Locus(D_1, P)
j = Segment(A, P)
c_1 = Ellipse(O, A, s / 2)
D = Point(c)
m = Segment(O, D)
E = Intersect(c_1, r, 2)
F = Intersect(c_1, r, 1)
n = Segment(E, F)
a = PathParameter(A)
G = Point(c)
t = Segment(D_1, P)
b = Segment(D_1, A)
op1 = false
h = 1 (slider from 0 to 5, step 0.02)
op2 = false
ec1:=XA - Xc = 0
h': 5 - h
c1 = Circle(O, s + h')
c2 = Circle(O, s - h')
sA = Circle(A, h')