# Proof of Hyperbolic Reflection

Let be a point on the hyperbola with foci and .
It must satisfy .
Choose point on such that .
Note that .
For midpoint of , the line is perpendicular to
and all four of the marked angles around are equal.
Let be any point on other than and connect it to , and .
because is a perpendicular bisector of .
by the triangle inequality.
Since , must lie outside the hyperbola,
so never passes through the hyperbola and must be tangent to it.
It follows that a ray will reflect off the hyperbola directly .
Similarly, a ray directed will reflect off the hyperbola .

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