Proof of Hyperbolic Reflection
Adapted from "Hyperbola: Proof of Reflective Property" by Brian Sterr
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 .