My book tells me the following:
Little Desargues Theorem: If two triangles are in perspective from a point $P$, and if two pairs of corresponding sides meet on a line $\ell$ through $P$, then the third pair of corresponding sides also meet on $\ell$.
As an exercise in the book, I'm asked to state and explain why the converse is true (supposedly, the converse should follow from the original statement itself).
My formulation of the converse is as follows: If two triangles are in perspective from a point $P$ and if one pair of corresponding sides meet on $\ell$, then the other two pairs of corresponding sides must meet on the same line $\ell$ through $P$.
Is my formulation correct? If so, I'm pretty confused as to why this follows from the original statement. Can someone please explain this exercise to me?
Your formulation of the converse of the Desargues’ Little Theorem is wrong, because what you have done is nothing but expressing the same theorem in a different way. A converse of a given theorem is a proposition whose premise and conclusion are the conclusion and premise of the given theorem.
Here is the correct formulation.
If two triangles are perspective from a line $\ell$, which is usually called the perspective axis, and if two straight lines joining the two of the three pairs of corresponding vertices of these two triangle meet on $\ell$ at a point $P$, then the line joining the remaining pair of vertices also intersects $\ell$ at $P$.
By the way, point $P$ is called the perspective center of the two triangles in question.
All the propositions in projective geometry occur in dual pairs which have the property that, starting from either proposition of a pair, the other can be immediately inferred by interchanging the parts played by the words $\mathbf{line}$ and $\mathbf{point}$. In your case, the given theorem, which is a very special case of the Desargues' theorem, states that some triangles are perspective from a line because they are perspective from a point. Then its converse must state that some triangles are perspective from a point because they are perspective from a line.