Proof of Hyperbolic Reflection

Let

be a point on the hyperbola

with foci

and

. It must satisfy

. Choose point

such that

. Note that

. For midpoint

, 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

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