Lenth of the tangent BD

ΔADB is Right Angle Triangle.

BD^{2} = AB^{2} - AD ^{2}

To draw a tangent to a circle from an external point

Tharanath

a = 6 (slider from 3 to 8, step 0.1)
A = (3.597756564753812, 0.4502155532659775)
c = Circle(A, a)
b = 9.2 (slider from 7 to 15, step 0.1)
B = Point(Circle[A, b])
d = Segment(A, B)
e = LineBisector(A, B)
C = Intersect(e, d)
f = Circle(C, B)
D = Intersect(c, f, 1)
E = Intersect(c, f, 2)
g = Ray(B, D)
h = Ray(B, E)
i = Ray(A, D)
j = Ray(A, E)
α = Angle(A, D, B)
β = Angle(B, E, A)
k = Segment(B, D)
l = Segment(B, E)
m = Segment(A, D)
n = Segment(A, E)
text3: "AD=" + m + ""
text4: "AB = " + d + ""
X: d^(2) - m^(2)
X1: sqrt(X)
text7: "BD = " + X1 + ""
text6: "BD^{2}=" + (LaTeX[d]) + "^2 - " + (LaTeX[m]) + "^2
              =(" + (LaTeX[d]) + "+" + (LaTeX[m]) + ")(" + (LaTeX[d]) + "-" + (LaTeX[m]) + ") =" + (LaTeX[X]) + ""