# Finding the length of side of trapezoid in two externally touching adjacent similar circles

The diagram shows two circles with centres $A$ and $B$ which touch at $C$ and have radius $r$.

The points $D$ and $E$, one on each circle, are such that $DE$ is parallel to the line $ACB$. Each of the angles $DAC$ and $EBC$ is $\theta$ radians where $0<\theta<\pi$.

Express the length of $DE$ in terms of $r$ and $\theta$.

I tried constructing a triangle drawing a line from $D$ to $C$. Used law of cosines to find $DC$ and compared the similarity of triangles $ADC$ and $DCE$, to find $DE$ but got $DE = 2r^2 - 2r^2cos\theta$ which is wrong. Hints, suggestions needed.

P.S. The length of DE at the back of book is given as $DE = 2r - 2r\cos\theta$.

Hint: drop perpendiculars from $D$ and $E$ onto $AB$. This yields a rectangle and two congruent triangles, all of which can be easily solved.