EqualBisectorsHyperbola

Equal bisectors property of rectangular hyperbolas

Consider an arbitrary point C on a rectangular hyperbola (c) and join it to the vertices A, B of (c) to form triangle ABC. Then the two bisectors of angle C (internal and external) of triangle ABC are parallel to the asymptotic lines of the hyperbola. In particular they are equal. Thus, we have another characterization of the rectangular hyperbola:

Given a fixed segment AB, the locus of points C such that the triangles ABC have equal internal and external bisectors at C, is the rectangular hyperbola with vertices at the points A and B.

In fact, from the discussion in RectangularAsProp.html follows that the asymptotic DG bisects angle(EDF), E, F being the middles of sides of ABC. But EDF being the medial triangle of ABC, the bisector of angle D is parallel to the bisector of angle C of ABC.

The same argument can be applied to the triangle resulting from ABC by changing the position of B on the hyperbola and taking A to be symmetric to B w.r. to D. This more general version is illustrated in the file BisectorRatio.html .

Equal bisectors property of rectangular hyperbolas

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