Three Circles with Two Common Tangents

point D

labels

colinear centers

second common tangent

Let

ABC

be a triangle circumscribed by the circle

γ

. Let

D

be a point on

AB

; form the line

CD

. Consider three other circles

γ

1

,

γ

2

, and

γ

3

with the common tangent

AB

, with

γ

1

inscribed in the triangle

ABC

, and

γ

2

and

γ

3

tangent to both the segment

CD

and

γ

. Prove that

γ

1

,

γ

2

, and

γ

3

have two common tangents.