The Double Butterfly Theorem on Hyperbolas

The butterfly theorem and the double butterfly theorem are fundamentally applied to circles. Subsequently, several authors have extended these theorems to ellipses and parabolas. As a hyperbola is another type of conic section, the author develops the double butterfly theorem for the hyperbola. In the proof process, the butterfly and double butterfly theorems for circles, ellipses, and parabolas rely on lemma of Haruki, which is applicable only to circles. Therefore, in this paper, the author first develops a version of lemma of Haruki specifically for the hyperbola. Based on this lemma, the proof of the double butterfly theorem for the hyperbola is then established. In other words, the double butterfly theorem can be extended to the hyperbola.