Draw your favorite obtuse angle with apex point $D$. Draw its bisector and choose a point $A$ on the bisector.
Drop an altitude $AE$ from the bisector to one of the legs of the original triangle. Draw the circle with center $A$ that touches the two legs of the angle. Its radius $AE$ will be smaller than $AD$.
Now draw a second circle also centered on $A$ with a radius _between_ $AE$ and $AD$. This circle has the legs of your obtuse angle as secant lines.
(If you wanted to start with a particular circle, you can now scale and translate the entire figure such that the last circle you drew coincides with the one you want).