On the fixed area property of the hyperbola (Q2798910)

The area of the triangle bounded by the asymptotes of a hyperbola and any of its tangent lines being fixed was known to Apollonius of Perga. In this paper the author proves that hyperbolas are essentially the only curves with this property. To prove this, the author establishes certain facts pertaining to the intermediate value property of certain functions, where a simple proof of Darboux's theorem on the intermediate value property for derivatives and several interesting questions and answers in elementary real analysis are demonstrated. The author also gives extensive remarks on the intermediate value property and continuity.