\mbox{point }E = sC + (1-s)D \longrightarrow \mbox{curve } g

\mbox{ intersecting points }P_1,P_2\mbox{ do not depend on s}

\mbox{If s > 1 and the ``locus disk''} \underline{g}\mbox{ is convex, then the ellipse c is contained in }\underline{g}

E' = sD + (1-s)C

C = \frac{s}{2s-1}E + (1-\frac{s}{2s-1})E'

a = 8 (slider from 0 to 10, step 0.5)
b = 4 (slider from 0 to 10, step 0.5)
r = 15 (slider from 0 to 20, step 0.5)
c: x^(2) / a^(2) + y^(2) / b^(2) = 1
text1: "c: \frac{x^2}{" + (LaTeX[a]) + "^2} + \frac{y^2}{" + (LaTeX[b]) + "^2} = 1"
theta = 0.5890486225480862 (slider from 0 to 6.283185307179586, step π / 16)
A: ((r * cos(theta)), (r * sin(theta)))
xhat(t) = (a * cos(t))
yhat(t) = (b * sin(t))
C = Point(c)
m = Line(A, C)
D = Intersect(c, m, 2)
s = 1.5 (slider from 0 to 5, step 0.25)
E: (s * C) + ((1 - s) * D)
k(t) = (yhat(t) - y(A)) / (xhat(t) - x(A))
a2(t) = ((a^(2) * k(t)^(2)) + b^(2)) / ((a^(2) * b^(2)))
a1(t) = (((2 * y(A)) * k(t)) - ((2 * x(A)) * k(t)^(2))) / b^(2)
x1(t) = ((-a1(t))) / a2(t) - xhat(t)
y1(t) = y(A) + (k(t) * (x1(t) - x(A)))
x_{E}(t) = (s * xhat(t)) + ((1 - s) * x1(t))
y_{E}(t) = (s * yhat(t)) + ((1 - s) * y1(t))
g = CurveCartesian(x_{E}(t), y_{E}(t), t, 0, (2 * π))
P_1: ((((a^(2) * y(A)) * sqrt((a^(2) * y(A)^(2)) + (b^(2) * x(A)^(2)) - (a^(2) * b^(2)))) + ((a^(2) * b^(2)) * x(A))) / ((a^(2) * y(A)^(2)) + (b^(2) * x(A)^(2))), ((-(((b^(2) * x(A)) * sqrt((a^(2) * y(A)^(2)) + (b^(2) * x(A)^(2)) - (a^(2) * b^(2)))) - ((a^(2) * b^(2)) * y(A))))) / ((a^(2) * y(A)^(2)) + (b^(2) * x(A)^(2))))
P_{2}: (((-(((a^(2) * y(A)) * sqrt((a^(2) * y(A)^(2)) + (b^(2) * x(A)^(2)) - (a^(2) * b^(2)))) - ((a^(2) * b^(2)) * x(A))))) / ((a^(2) * y(A)^(2)) + (b^(2) * x(A)^(2))), (((b^(2) * x(A)) * sqrt((a^(2) * y(A)^(2)) + (b^(2) * x(A)^(2)) - (a^(2) * b^(2)))) + ((a^(2) * b^(2)) * y(A))) / ((a^(2) * y(A)^(2)) + (b^(2) * x(A)^(2))))
Q_1: (25 * (x(P_1) / a^(2), y(P_1) / b^(2)))
Q2: (10 * (x(P_{2}) / a^(2), y(P_{2}) / b^(2)))
grad_{c,P1} = Vector(P_1, P_1 + Q_1)
vec_{P1A} = Vector(P_1, A)
α = Angle(grad_{c,P1}, vec_{P1A})
E': (s * D) + ((1 - s) * C)