Since no respondents provided a solution to his last problem, Prof. Aliyev is willing to offer one more chance to unravel his mysterious diagrams. In the accompanying figure, nonintersecting circles with radii
r
<
R
have common
external tangent lines
A
B
and
C
D
, where the named points are
the points of tangency. Choose point
M
on segment
A
B
so that
angles
A
M
D
and
B
M
C
are equal. Let
θ
be the angle
D
C
M
.
Show that
sin
θ
≤
r
/
R
.