Hello! 
Let $G=A\underset{C}\star B$ be an amalgamated Product. 
Let $a\in A$. If a is conjugated to an Element $b\in B$, then $a$ is conjugated to an Element $c\in C$. The Question is: Why is that true? It is clear, when $a\in A\cap C=C$. 
It seems to be very easy. But at the moment, i think, i make a fault while im calculating with the elements. 



Conjugated elements in amalgameted Product
There is also a simple geometric proof using Bass-Serre theory, which does not require much in the way of calculation. Let $G$ act on the Bass-Serre tree $T$ of the amalgamated free product $A\*\_CB$. The stabilizer of every edge of $T$ is conjugate to $C$, and the vertices of the tree $T$ are partitioned into two subsets $V\_A$, $V\_B$ so that the stabilizer of every vertex in $V\_A$ is conjugate to $A$ and the stabilizer of every vertex in $V\_B$ is conjugate to $B$. By your assumption, the element $a$ fixes some vertex in $V\_A$ and some vertex in $V\_B$, so $a$ fixes the edge path between those two vertices, so $a$ fixes some edge along that edge path, so $a$ is conjugate to an element of $C$.


