The question is as follows:
Reflection property of the ellipse. Suppose that $P$ is a point on an ellipse whose focal points are $F_1$ and $F_2$. Draw the intersecting lines $PF_1$ and $PF_2$, as well as the bisectors of the four angles they form. Consider the bisector that does not separate $F_1$ and $F_2$. Prove that given any point $Q$ other than $P$ on this line, $QF_1 + QF_2 > PF_1 + PF_2$. Explain why the line meets the ellipse only at $P$. Justify the title of this problem.
I drew the angle bisectors of the four angles that are formed after intersecting the lines extending from $PF_1$ and $PF_2$. It looks like from the drawing that the bisector that is not separating the foci is the line tangent to the ellipse at $P$. I, however, can't prove that that statement is true. Any help will be greatly appreciated.