BisectorRatio

Bisector ratio constancy property of rectangular hyperbolas

Consider an arbitrary point C on a rectangular hyperbola (c) and join it to two points A, B of (c), symmetric w.r. to the center D of (c), to form triangle ABC. Then the two bisectors u, v of angle C (internal and external) of triangle ABC are parallel to the asymptotic lines of the hyperbola. In particular their ratio u/v is constant and independent of the position of C on (c). 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 the ratio of the internal to the external bisector at C equal to a constant k, is a rectangular hyperbola passing through the points A and B. A particular case of this property is discussed in EqualBisectorsHyperbola.html .

In fact, from the discussion in RectangularAsProp.html follows that asymptotic 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.

Notice that the axes and the asymptotic lines, passing through the middle of AB can be constructed from the data of the triangle ABC. Then the vertex I of the hyperbola can be determined from an asymptotic triangle DD'D'' of the hyperbola whose area e satisfies e = DI².
To construct an asymptotic triangle of the hyperbola project A on the asymptotics to A', A'' and take the symmetrics D', D'' of D w.r. to these points. See the analysis of this idea in the file HyperbolaRectangular.html .

Bisector ratio constancy property of rectangular hyperbolas

See Also