Converses of statements Does this statement have a converse?
Denote by $A_2$ the circumcenter of $\triangle{B_1C_1D_1}$, and define $B_2, C_2,D_2$ in an analogous way. Show that the quadrilateral $A_2B_2C_2D_2$ is similar to the quadrilateral $ABCD.$
I think it does since we can equivalently write "If we denote by $A_2$ the circumcenter of $\triangle{B_1C_1D_1}$, and define $B_2, C_2,D_2$ in an analogous way, then the quadrilateral $A_2B_2C_2D_2$ is similar to the quadrilateral $ABCD.$"
But the converse is obviously false. This leads me to the question of to find the converse of a statement, do we logically reduce it to something else as I have done above?
 A: The converse of a statement is not logically equivalent to the original statement. So if your original statement is true, it does not imply that its converse is true.
To examine the converse of a statement it needs to be of the following form: 
$$ \text{If}\,\, P, \text{then} \,\,Q.$$
A converse is only defined for implications of this form.
So your statement would first need to be rephrased into an implication which you correctly did: "If $A_2$ denotes $\triangle B_1C_1D_1$ and $B_2,C_2,D_2$ are defined in an analogous way, then $A_2B_2C_2D_2$ is similar to the equilateral $ABCD$."  
The converse then takes the form 
$$ \text{If}\,\,Q, \text{then} \,\,P.$$
So the converse of your statement is now: "If $A_2B_2C_2D_2$ is similar to the equilateral $ABCD$, then $A_2$ denotes $\triangle B_1C_1D_1$ and $B_2,C_2,D_2$ are defined in an analogous way." 
This resulting statement may or may not be true, but we cannot determine its truth based only on our original statement. So it is perfectly ok that a true statement has a false converse.
