Suppose I have a unit square, and I mark some number of points in it with my pen wherever I like. Then I mark a final point, and I must place this final point at the center of the largest possible circle that can be drawn without any of the other points in its interior - in other words, I place this last point as far from the edges or other points as possible. Then, my friend who was waiting in the other room comes and looks at all the points I placed and tries to guess which one was the final point placed. It's my win if my friend cannot deduce which point was the final one, and it's my loss if my friend is able to deduce with certainty which point was the final one.
For example, suppose I first place only a single point at the coordinates $(0.1, 0.1)$. Then the final point must go in the exact center of the square. My friend sees a point at $(0.1, 0.1)$ and at $(0.5, 0.5)$. He knows that if the $(0.5, 0.5)$ point had been placed first that the second point would have to be placed at $(0.29289..., 0.29289...)$, $(0.29289..., 0.70710...)$, $(0.70710..., 0.29289...)$, or $(0.70710..., 0.70170...)$; this tells him that $(0.5, 0.5)$ could not have been placed first, so I have lost the game. Suppose instead I first place my first point only at $(R, R)$ where $R = \frac{4-\sqrt{2}}{7}$. Then the final point is placed exactly opposite of it at $(1-R, 1-R)$, and my friend cannot know which of the two points is which. The same is true if I had placed my point at $(R, R)$, $(R, 1-R)$, $(1-R, R)$, or $(1-R, 1-R)$. Placing a single point at one of these four positions will win the game, and placing a single point elsewhere will lose; this characterizes every case in which I initially place just a single point.
My question is this: if I initially place a pair of points (followed by a final, third point): which pairs of points allow my friend to be completely sure as to which was the final point, which pairs of points allow my friend to be sure of one point that is not the final point but unsure which of the two others is, and which pairs of points make my friend entirely uncertain as to which of the three resultant points came last?