I need to find $\pi_1(X,x_0)$ where $X$ is the intersection of two circles (in $\mathbb{R^2}$). My idea is to use van Kampen, with $A,B,A\cap B$ as in figure. Both $A$ and $B$ are homotopic equivalent to $S^1 \vee S^1$ (firstly I deformation retract $A$ to $S^1$ united a diameter, then I can contract the diameter to a point). $A\cap B$ is homotopic equivalent to $S^1$. Then $\pi_1(X,x_0)=\pi_1(A,x_0)*_{\pi_1(A\cap B,x_0)}\pi_1(B,x_0)=(\mathbb{Z}*\mathbb{Z})*_{\mathbb{Z}} (\mathbb{Z}*\mathbb{Z})$.I think the result should be $\mathbb{Z}*\mathbb{Z}*\mathbb{Z}$ but I don't know how to prove it. Any suggestion?
Collaps one of the (short) segments connecting the two intersection points to a single point. This map is a homotopy equivalence. The resulting space is $S^1 \vee S^1 \vee S^1$ whose fundamental group is $\mathbb{Z}*\mathbb{Z}*\mathbb{Z}$.