I am writing a report and as part of it I need to prove the property that for a point $P$ on a hyperbola, the tangent to the hyperbola at $P$ bisects the angle $\angle F_1PF_2$ where $F_1$ and $F_2$ are the foci of the hyperbola.

Wikipedia has the following proof which is pretty neat but obviously doesn't constitute an academic reference: https://en.wikipedia.org/wiki/Hyperbola#The_tangent_bisects_the_angle_between_the_lines_to_the_foci

Does anybody know of any books or papers which use the same proof?

  • 1
    $\begingroup$ Have you tried looking at the references given in the Wikipedia article? $\endgroup$
    – gmvh
    Apr 18, 2021 at 14:55

1 Answer 1


This is the "reflection property" of the hyperbola. In the context of "academic references", it is good practice to cite the original source, which is

Apollonius of Perga, 200 BCE, Treatise on Conic Sections, book III, proposition 48; English translation by T.L. Heath, page 116 (Cambridge University Press, 1896).

Alternatively, you may prefer a more recent proof, so here is one:

• Michael K. Brozinsky, Reflection Property of the Ellipse and the Hyperbola, The College Mathematics Journal, Vol. 15, No. 2 (1984), pp. 140-142. https://doi.org/10.2307/2686519


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