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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 from focus will reflect off the hyperbola directly away from focus . Similarly, a ray directed to focus will reflect off the hyperbola toward focus .