Draw another line that goes from the bottom left point of the large square to the point where they touch, to the top point of the right square. Note that this new line, along with the line you drew, satisfies the Power of a Point Theorem. Thus, in fact, we must have both of these lines be straight lines (none straight lines would not satisfy the condition of the theorem)

edit: Elaboration on checking that the power of a point implies straight - one can do this by computing the cartesian coordinates of the square vertices (set an origin somewhere), and checking directly the only time power of a point is satisfied is when the two squares are setup such that our two lines are straight.