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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.

So in my proof I've shown that two angle bisectors are perpendicular to each other. I've also proven that that $PF_1 + PF_2 = F_1F_2'$ where $F_2'$ is a reflection of $F_2$ across the line that is tangent to the ellipse and and contains point P.

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Take any other point $Q\ne P$ on the line and show that $$ QF_1+QF_2=QF_1+QF'_2>F_1F'_2, $$ using triangle inequality.

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