$A(0,12)$

$B(-5,0)$

$C(5,0)$

$Step=%v$

$∵ \frac{r_1}{12-r_1}=\frac{5}{13} $

$ ∴13\;r_1=60-5\;r_1 $

$⇒ r_1=\frac{60}{18}=\frac{10}{3} $

$∵ \frac{r_2}{12-2r_1-r_2}=\frac{5}{13} $

$ ∴13\;r_2=60-10×\frac{10}{3}-5\;r_2 $

$ ⇒18\;r_2 =\frac{80}{3}⇒r_2=\frac{40}{27}$

$ ⇒r=\frac{\pi\;r_2^2}{\pi\;r_1^2}=\left( \frac{\frac{40}{27}}{\frac{10}{3}}  \right) ^2=\frac{16}{81}$

$  ⇒s=\frac{\pi×\frac{100}{9}}{1-\frac{16}{81}}=\frac{180\;\pi}{13}$

$ 無窮等比級數示例 $

A = (0, 12)
B = (-5, 0)
C = (5, 0)
a = Segment(B, C)
b = Segment(A, C)
c = Segment(A, B)
d, e = AngularBisector(c, a)
f, g = AngularBisector(b, a)
O_1 = Intersect(e, f)
c_1 = Circle(O_1, Distance[O_1, a])
D_1 = Intersect(c_1, b)
r_1 = Segment(O_1, D_1)
O = (0, 0)
O' = Mirror(O, O_1)
i = Line(O', a)
j, k = AngularBisector(c, i)
l, m = AngularBisector(b, i)
O_2 = Intersect(l, k)
c_2 = Circle(O_2, Distance[O_2, O'])
n = Line(O_2, r_1)
D_2 = Intersect(b, n)
r_2 = Segment(O_2, D_2)
O'' = Mirror(O', O_2)
q = Line(O'', a)
r, s_1 = AngularBisector(c, q)
t, a_1 = AngularBisector(b, q)
O_3 = Intersect(s_1, t)
c_3 = Circle(O_3, Distance[O_3, O''])
b_1 = Line(O_3, r_2)
D_3 = Intersect(b_1, b)
r_3 = Segment(O_3, D_3)
O''' = Mirror(O'', O_3)
e_1 = Line(O''', a)
f_1, g_1 = AngularBisector(e_1, c)
h_1, i_1 = AngularBisector(e_1, b)
O_4 = Intersect(g_1, h_1)
c_4 = Circle(O_4, Distance[O_4, O'''])
j_1 = Line(O_4, r_3)
D_4 = Intersect(b, j_1)
r_4 = Segment(O_4, D_4)
s = 0 (slider from 0 to 4, step 1)
tri1, c_5, o, a_2 = Polygon(O, A, C)
tri2, d_1, o_1, a_3 = Polygon(O_1, A, D_1)
tri3, d_2, o_2, a_4 = Polygon(O_2, A, D_2)